This appeal requires us to sort through a complicated set of commercial machinations and evaluate the soundness of an equitable accounting through which the lower court divided the surplus proceeds of a multi-million-dollar foreclosure sale. To solve this conundrum, we must answer three loosely related questions. The first concerns the applicability of the doctrine of judicial estoppel, the second concerns the way in which the methodology for calculating the equity of redemption fits within the framework of a judicial accounting, and the third concerns the propriety of a post-trial joinder of additional defendants. After careful consideration of these questions, we reject the district court’s proposed application of the doctrine of judicial estoppel but uphold its treatment of the equity of redemption and its joinder order. When all is said and done, we affirm in part, reverse in part, vacate the judgment, and remand for further proceedings consistent with this opinion.
I. BACKGROUND
Although there is a long, convoluted, and sometimes Machiavellian history involving the protagonists, we relate here only those facts relevant to the issues presented on appeal. We supplement this account in connection with our discussion of particular issues. Throughout, we accept the district court’s factual findings to the extent that they are not clearly erroneous. Limone v. United States, 579 F.3d 79, 94 (1st Cir.2009); Cumpiano v. Banco Santander P.R., 902 F.2d 148, 152 (1st Cir.1990); Fed.R.Civ.P. 52(a).
Michael Perry and Stephen Yellin each hold a fifty percent interest in Condominium Housing, Incorporated (CHI), which owned a large apartment complex in Boston, Massachusetts, known as “the Fen-more.” CHI purchased the Fenmore from Harold Brown in 1985. The purchase price included two promissory notes, with an aggregate face value of approximately $11,000,000 (the Notes), executed by Perry and Yellin as co-makers. The Notes were secured by first and second mortgages on the property.
Over time, Perry and Yellin made payments on the Notes. But when the Boston real estate market cratered in the late 1980s, they defaulted on several obligations, including not only the Notes but also an array of loans from their primary lender, Capitol Bank (which, among other things, held a third mortgage on the Fen-more). By 1990, Perry and Yellin owed Capitol Bank more than $7,000,000. They negotiated a settlement with the bank in July of that year, but the settlement proved to be illusory. The bank subsequently repudiated it, and Perry and Yellin were forced to sue for specific performance in a Massachusetts state court.
*6During the currency of that suit, the Federal Deposit Insurance Corporation (FDIC) took over Capitol Bank as its receiver and liquidating agent. The FDIC sought to collect Perry’s and Yellin’s indebtedness to the bank, claiming that they owed roughly $19,000,000 in principal and accrued interest on various loans. As part of its collection effort, the FDIC disavowed the earlier settlement and, in January of 1999, filed an amended counterclaim in the state court suit.
The amended counterclaim named as defendants Perry, Yellin, and a number of their relatives. These additional defendants (whom we shall call the “Perry Parties” and the “Yellin Parties”) were allegedly involved in Perry’s and Yellin’s real estate enterprises as “straws.” In due season, the FDIC removed the state court action to the federal district court. See 12 U.S.C. § 1819(b)(2)(B).
Brown’s fortunes also had been adversely affected by the slumping real estate market. In 1991, he filed for bankruptcy. The bankruptcy proceedings dragged on and, in August of 1996, he submitted an affidavit to the bankruptcy court in which he represented that the Notes had an unpaid balance of $902,662 and were uncol-lectible. The bankruptcy court granted Brown a discharge from bankruptcy in September of 1996 and permitted him to retain ownership of the Notes.
CHI went into bankruptcy in April of 1996. After Brown emerged from his own bankruptcy, he intervened in CHI’s bankruptcy and requested relief from the automatic stay, 11 U.S.C. § 362, so that he could foreclose on the Fenmore. Brown represented that the Notes had a principal balance of $902,662 and past-due interest of $950,620. The bankruptcy court granted Brown’s motion to lift the stay in December of 1996. Instead of foreclosing, however, Brown agreed to sell the Notes to Yellin for $950,000. Yellin effectuated this purchase behind Perry’s back and through a straw: Steven Blum, in his capacity as trustee of Moorings Nominee Trust. The sole beneficiary of the trust was North Shore Renewal, Inc., a shell corporation wholly owned by Yellin’s wife, Elaine. Yellin, acting through Blum, then took control of the Fenmore as a mortgagee-in-possession and began collecting rents.
In June of 1997, Perry, on behalf of CHI, sued the Yellins and Blum, individually and as trustee of Moorings Nominee Trust, in a Massachusetts state court. The suit alleged an alphabet of wrongdoing, including breach of fiduciary duty, fraud, and conversion, stemming from the purchase of the Notes. In October of 1998, after Perry failed to attend a pretrial conference, the state court dismissed the suit for want of prosecution. CHI v. Blum, No. 97-3007 (Mass.Dist.Ct. Oct. 7, 1998) (unpublished order).
In late 1997, Yellin, acting through Blum, commenced foreclosure proceedings with respect to the Fenmore. The FDIC, which held junior mortgages on the property as the receiver for Capitol Bank, responded by bringing an action in the federal district court. In the action, the FDIC sought to enjoin any foreclosure sale. The district court granted a preliminary injunction blocking the foreclosure sale. Later, it consolidated the action in which it had granted the injunction with the action that the FDIC had removed from the state court.
In 2002, Yellin and the Yellin Parties settled with the FDIC for $5,000,000. Under the terms of the settlement, the FDIC permitted Yellin, acting through Blum, to foreclose on the Fenmore and use the first $5,000,000 of the foreclosure proceeds to fund the settlement. The district court thereafter dissolved the existing injunction *7and Blum foreclosed on the Fenmore. The apartment complex was sold at auction for $9,450,000 (ironically, to Brown). The FDIC and the Yellin Parties then jointly moved to dismiss all claims inter sese. The district court granted that motion on June 10, 2002.1
Following the foreclosure sale, Perry, on behalf of himself and CHI, cross-claimed against Blum for an accounting of both the foreclosure proceeds and all rents collected between 1996 and 2002. Perry alleged that, as an equal partner in CHI, he was entitled to one-half of the Fenmore’s equity of redemption.
In February and March of 2005, the district court held a bench trial on the cross-claim. The Notes were secured by first and second mortgages on the Fen-more, so Blum, as the noteholder and as a straw for Yellin, had a priority claim to the foreclosure proceeds. Not surprisingly, then, one of the main issues at trial concerned the amount due on the Notes. Blum contended that $7,494,435 was due. Perry contended that the amount due was much less. The district court determined that Blum was estopped from asserting that the amount due was anything other than what Brown, Blum’s predecessor-in-interest, had represented in the bankruptcy court, namely, $1,853,282. Perry v. Blum, (Perry I), No. 99-12194, slip op. at 30 (D.Mass. Oct. 31, 2008).2 Even though the trial had ended years earlier, the court granted Perry’s motion to join the Yellins as reach-and-apply defendants. Id., slip op. at 36.
Before the court actually made a final accounting, Blum and the Yellins moved for reconsideration of Perry I, arguing that the court had erred in applying judicial estoppel and in joining the Yellins after trial. The court denied the motion, Perry v. Blum (Perry II), No. 99-12194, slip op. at 2 (D. Mass. June 2, 2009). The court then prepared the accounting and entered judgment for Perry against Blum and the Yellins in the sum of $4,347,126, plus pre-judgment interest. The appendix to this opinion delineates the manner in which the court calculated this amount.
Blum and the Yellins moved to alter or amend the judgment, Fed.R.Civ.P. 59(e), protesting that the court had erred in excluding from its calculation of the equity of redemption an offset for the $5,000,000 payment that the Yellins had made to the FDIC — an offset that would have had the effect of reducing Perry’s award by $2,500,000. The district court denied the motion, reasoning that it would be inequitable to allow Yellin to settle his personal debts by using foreclosure proceeds that otherwise would have to be split with Perry. Perry v. Blum (Perry III), No. 99-12194 (D. Mass. June 18, 2009) (unpublished order). Blum and the Yellins filed a timely motion of appeal from both the final judgment and the denial of their Rule 59(e) motion.
II. ANALYSIS
Like a milking stool, this appeal rests on three legs. The appellants (Blum and the Yellins) insist that the district court erred in (i) invoking judicial estoppel, (ii) miscalculating the equity of redemption, and (iii) adding the Yellins as defendants after the trial had ended. We discuss each of these *8arguments separately. Before doing so, however, we pause to make a point about choice of law.
Due to the FDIC’s involvement, this case had its roots in federal question jurisdiction. 12 U.S.C. § 1819(b)(2)(A); 28 U.S.C. § 1331. But the FDIC has departed from the scene, and the remaining claims are cognizable only under supplemental jurisdiction. 28 U.S.C. § 1367. Where, as here, a federal court proceeds under supplemental jurisdiction, it is obliged to apply federal procedural law and state substantive law. Hoyos v. Telecorp Commc’ns, Inc., 488 F.3d 1, 5 (1st Cir.2007). To the extent that procedural issues loom, the Federal Rules of Civil Procedure provide the beacon by which we must steer.
Most of the components of the appellants’ asseverational array raise substantive questions1 and, thus, require us to apply Massachusetts law. Withal, one of the issues presents a thorny problem of classification, which this court has not resolved: Is judicial estoppel procedural (and, thus, governed by federal law) or substantive (and, thus, governed by state law)?
This is an interesting Rubik’s cube, but we need not provide a definitive answer here. The parties and the district court all assumed that federal standards of judicial estoppel governed, and the case has been briefed and argued on the same assumption. In such circumstances, we may hold the parties to their plausible choice of law, whether or not that choice is correct. Sec Thore v. Howe, 466 F.3d 173, 181 n. 1 (1st Cir.2006); Alt. Sys. Concepts, Inc. v. Synopsys, Inc., 374 F.3d 23, 32 (1st Cir.2004). We follow that prudential course and apply federal law in discussing this issue.
A. Judicial Estoppel.
The appellants insist that the district court erred in invoking judicial estop-pel to preclude them from showing that the amount due on the Notes was anything other than what Brown had represented in a prior bankruptcy proceeding. We review for abuse of discretion a district court’s application of judicial estoppel. Global NAPs, Inc. v. Verizon New Engl. Inc., 603 F.3d 71, 91 (1st Cir.2010). Within that rubric, we accept the trial court’s findings of fact unless they are clearly erroneous, see Limone, 579 F.3d at 94; Cumpiano, 902 F.2d at 152, and evaluate its answers to abstract questions of law de novo, see San Juan Cable LLC v. P.R. Tel. Co., 612 F.3d 25, 29 (1st Cir.2010). We treat a material mistake of law as a per se abuse of discretion. Rosario-Urdaz v. Rivera-Hernandez, 350 F.3d 219, 221 (1st Cir.2003).
The doctrine of judicial estoppel is equitable in nature. It operates to prevent a litigant from taking a litigation position that is inconsistent with a litigation position successfully asserted by him in an earlier phase of the same case or in an earlier court proceeding. InterGen N.V. v. Grina, 344 F.3d 134, 144 (1st Cir.2003). The purpose of the doctrine is to protect the integrity of the judicial process. It is typically invoked when a litigant tries to play fast and loose with the courts. New Hampshire v. Maine, 532 U.S. 742, 749-50, 121 S.Ct. 1808, 149 L.Ed.2d 968 (2001); Alt. Sys. Concepts, 374 F.3d at 33; Patriot Cinemas, Inc. v. Gen. Cinemas Corp., 834 F.2d 208, 212 (1st Cir.1987).
The contours of judicial estoppel are hazy. But even though its elements cannot be reduced to a scientifically precise formula, New Hampshire, 532 U.S. at 750, 121 S.Ct. 1808, courts generally require the presence of three things before introducing the doctrine into a particular *9ease. First, a party’s earlier and later positions must be clearly inconsistent. Id.) Alt. Sys. Concepts, 374 F.3d at 33. Second, the party must have succeeded in persuading a court to accept the earlier position. New Hampshire, 532 U.S. at 750, 121 S.Ct. 1808; Alt. Sys. Concepts, 374 F.3d at 33. Third, the party seeking to assert the inconsistent position must stand to derive an unfair advantage if the new position is accepted by the court. New Hampshire, 532 U.S. at 751, 121 S.Ct. 1808; Alt. Sys. Concepts, 374 F.3d at 33.
Ordinarily, the party against whom judicial estoppel is invoked must be the same party who made the prior (inconsistent) representation. See InterGen, 344 F.3d at 144 (explaining that judicial estop-pel “prevents a litigant from pressing a claim that is inconsistent with a position taken by that litigant” in the same or an earlier proceeding); Brewer v. Madigan, 945 F.2d 449, 455 (1st Cir.1991) (explaining that judicial estoppel prevents “a party from taking a position inconsistent with one successfully and unequivocally asserted by that same party in a prior proceeding”). Courts normally refuse to apply judicial estoppel to one party based on the representations of an unrelated party. See, e.g., Parker v. Wendy’s Int’l, Inc., 365 F.3d 1268, 1272 (11th Cir.2004); Bethesda Lutheran Homes & Servs., Inc. v. Born, 238 F.3d 853, 858 (7th Cir.2001); Tenn. ex rel. Sizemore v. Surety Bank, 200 F.3d 373, 381-82 (5th Cir.2000); see also 18B Charles A. Wright, Arthur R. Miller & Edward H. Cooper, Federal Practice and Procedure § 4477, at 618-19 (2d ed. 2002). Nevertheless, courts sometimes have allowed judicial estoppel when the estopped party was responsible in fact for the earlier representation, see, e.g., Ladd v. ITT Corp., 148 F.3d 753, 756 (7th Cir.1998), or when the estopped party was the assignee of a litigation claim or assumed the original party’s role, see 18B Wright et al., supra, § 4477, at 618-19.
In the case at hand, the district court determined that the appellants (Blum and the Yellins) should be judicially estopped from asserting that the amount due on the Notes was $7,494,435 because a third party, Brown, had represented in his own bankruptcy proceeding that the unpaid balance was a mere $902,662, and had represented in CHI’s bankruptcy that the total amount due on the Notes (including accrued interest) was $1,853,282. Perry I, slip op. at 30. The court bound the appellants to Brown’s earlier representations primarily on the theory that the appellants, as purchasers of the Notes, were subject to any defenses that Perry and CHI could have asserted against Brown himself. Id. at 31 & n. 15.3 The court reasoned that because Brown would be judicially estopped from contradicting his prior representations anent the amount due on the Notes, his assignees should be similarly estopped. Id. at 32-36. The appellants say that Brown’s representations cannot be imputed to them and that, therefore, judicial estoppel was incorrectly invoked.
*10In examining these competing contentions, we take the underlying substantive law — the law of negotiable instruments— from Massachusetts. Under Massachusetts law, negotiable instruments, such as the Notes, are governed by Article III of the Uniform Commercial Code (U.C.C.). See Mass. Gen. Laws ch. 106, § 3-102. The U.C.C. states that a noteholder’s right to enforce a note is subject to certain defenses. Id. § 3-305(a). These defenses include those enumerated in section 3-305(a)(1) (often termed “real defenses”), defenses specifically listed elsewhere in Article III, and defenses “that would be available if the person entitled to enforce the instrument were enforcing a right to payment under a simple contract” (often termed “personal defenses”). Id. If a noteholder qualifies as a holder in due course, his right to enforce the note is subject only to real defenses. Id. § 3-305(b).
The appellants do not presume to be holders in due course. See Perry II, slip op. at 12. Thus, they are subject to the axiom that the rights of a transferee who is not a holder in due course rise no higher than the rights of the transferor. See 2 James J. White & Robert S. Summers, Uniform Commercial Code § 17-11, at 226 (5th ed. 2008). It follows that the appellants, as transferees, are subject to the trilogy of defenses described in section 3-305(a) to the same extent that those defenses would have been available against the transferor (Brown). See id.; see also 25 Herbert Lemelman, Massachusetts Practice § 3:150, at 428 (3d ed. 2002) (stating in essence that a holder who is not a holder in due course is treated under Massachusetts law as the assignee of a contract).
This does not mean, however, that the transferor and the transferee are to be treated as one and the same for all purposes. Of particular pertinence for present purposes, judicial estoppel does not fit comfortably within any of the trilogy of defenses described in section 3-305(a). Judicial estoppel is certainly not a “real defense” within the provision of the statute, nor is it a defense specifically listed anywhere in Article III of the U.C.C. This leaves only the category of “personal defenses.”
Personal defenses typically are thought to be “those based on common law contract principles.” Mass. Gen. Laws ch. 106, § 3-305(a) emt. 2; see FDIC v. Wood, 758 F.2d 156, 160 (6th Cir.1985) (“ ‘Personal’ defenses, such as failure of consideration and usury ... are defenses or claims stemming from the underlying transaction.”). Judicial estoppel does not fit seamlessly into the taxonomy of personal defenses, as it is not a defense aimed directly at either the validity or the enforceability of a contract. Rather, it is a judge-made doctrine designed to protect the integrity of the judicial system. See New Hampshire, 532 U.S. at 749, 121 S.Ct. 1808. This seeming incongruence gives us some pause about trying to ram the square peg of judicial estoppel into the round hole of personal defenses.
Still, we recognize that the boundaries of judicial estoppel are hazy. It is conceivable that cases may arise in which the doctrine can be considered a personal defense, used to prevent a noteholder from playing fast and loose with the courts through, say, engaging a straw to assert a payoff amount contradictory to one earlier put forth by the noteholder himself. See G-I Holdings, Inc. v. Reliance Ins. Co., 586 F.3d 247, 262 (3d Cir.2009) (“We will aPPly [judicial estoppel] to neutralize threats to judicial integrity however they may arise.”); cf. FDIC v. Gulf Life Ins. Co., 737 F.2d 1513, 1518 (11th Cir.1984) (suggesting that, for negotiable instru*11ments, general estoppel may constitute a personal defense, “to which a holder in due course would be impregnable”). The combinations of possible circumstances are infinitely varied, and a flat holding that judicial estoppel can be circumvented simply by transferring property to a third party would be folly. See generally Sandstrom v. ChemLawn Corp., 904 F.2d 83, 87-88 (1st Cir.1990) (emphasizing utility of judicial estoppel as a means of preventing a party from obtaining an unfair advantage).
Here, however, we need not grapple with the admittedly difficult question of whether judicial estoppel may ever qualify as a personal defense. In our view, the district court’s application of judicial estop-pel suffers from a different infirmity: Perry has failed to satisfy the second requirement for judicial estoppel. He has not shown that the bankruptcy court actually accepted Brown’s representation of the value of the Notes.
The party proposing an application of judicial estoppel must show that the relevant court actually accepted the other party’s earlier representation. See Gens v. Resolution Trust Corp., 112 F.3d 569, 572 (1st Cir.1997) (“Judicial estoppel is not implicated unless the first forum accepted the legal or factual assertion alleged to be at odds with the position advanced in the current forum.... ” (emphasis in original)). “Acceptance” in this context is a term of art. In order to satisfy this prerequisite, a party need not show that the earlier representation led to a favorable ruling on the merits of the proceeding in which it was made, but must show that the court adopted and relied on the represented position either in a preliminary matter or as part of a final disposition. See, e.g., Pennycuff v. Fentress Cnty. Bd. of Educ., 404 F.3d 447, 453 (6th Cir.2005); Karaha Bodas Co. v. Perusahaan Pertambangan Minyak Dan Gas Bumi Negara, 364 F.3d 274, 294 (5th Cir.2004); see also Global NAPs, 603 F.3d at 90 (finding that court accepted party’s first representation by relying on it in granting temporary restraining order); Alt. Sys. Concepts, 374 F.3d at 34 (explaining that because court relied on party’s initial position in denying motion to dismiss, party “derived a direct (if temporary) benefit from its original position”).
The showing of judicial acceptance must be a strong one. See SBT Holdings, LLC v. Town of Westminster, 547 F.3d 28, 37 & n. 8 (1st Cir.2008) (rejecting judicial estoppel argument because record was unclear as to whether court accepted plaintiffs prior position); cf. United States v. Pakala, 568 F.3d 47, 60 (1st Cir.2009) (upholding judicial estoppel when it was “clearly obvious” that the original trial court, in granting a motion, “necessarily adopted” the position that the movant later sought to contradict); United Nat’l Ins. Co. v. Spectrum Worldwide, Inc., 555 F.3d 772, 779 (9th Cir.2009) (authorizing judicial estoppel when trial court “clearly accepted and relied upon [the party’s] assertions” when it denied preliminary injunction). The need for a strong showing derives from the maxim that “[jjudicial estoppel is applied with caution to avoid impinging on the truth-seeking function of the court because the doctrine precludes a contradictory position without examining the truth of either statement.” Teledyne Indus., Inc. v. NLRB, 911 F.2d 1214, 1218 (6th Cir.1990); accord Lowery v. Stovall, 92 F.3d 219, 224 (4th Cir.1996) (“The insistence upon a court having accepted the party’s prior inconsistent position ensures that judicial estoppel is applied in the narrowest of circumstances.”).
It follows that a proponent of judicial estoppel must affirmatively show, by competent evidence or inescapable inference, that the prior court adopted or relied *12upon the previous inconsistent assertion. See United Steelworkers of Am. v. Ret. Income Plan for Hourly-Rated Emps. of ASARCO, Inc., 512 F.3d 555, 563-64 (9th Cir.2008) (finding that failure to demonstrate judicial acceptance precludes application of estoppel where “the district court never held ” the notion urged by the party) (emphasis in original); United States v. Levasseur, 846 F.2d 786, 794 (1st Cir.1988) (rejecting “illogical surmise” about what prior court might have accepted as a basis for judicial estoppel). Perry has not made the requisite showing here.
There are only two possible actions taken by the bankruptcy court that Brown’s representation might have affected. First, in Brown’s bankruptcy proceeding, the court allowed him to retain the Notes after he stated the amount due and represented that the Notes were uncollectible. Second, in the CHI bankruptcy (in which Brown intervened), the court lifted the automatic stay, allowing Brown to foreclose on the Fenmore. There is no evidence that either action was premised on Brown’s statements about the amount owed on the Notes, and it is implausible to infer any such nexus.
We examine these two actions in reverse order. It defies logic to deduce that the bankruptcy court granted Brown relief from the automatic stay on the basis of the amount owed on the Notes. A larger figure would not have supported keeping the stay in place — if anything, a greater amount owed would have created an additional incentive for the court to allow the foreclosure to proceed post-haste.
Similarly, it is utterly speculative to suggest that the bankruptcy court, in approving Brown’s global settlement and allowing him to retain the Notes, accepted each and every one of his figures. Generally speaking, settlement “neither requires nor implies any judicial endorsement of either party’s claims or theories.” In re Bankvest Capital Corp., 375 F.3d 51, 60 (1st Cir.2004) (quoting Bates v. Long Island R.R. Co., 997 F.2d 1028, 1038 (2d Cir.1993)). So viewed, an unexplained settlement does not provide the prior success necessary for judicial estoppel. See C & M Props., LLC v. Burbidge, 377 B.R. 677, 685 (D.Utah 2007) (noting that “[t]he the fact that [debtor] obtained confirmation of its [plan] does not demonstrate that the bankruptcy court was misled” and thus finding that second element of judicial es-toppel was not satisfied), vacated on other grounds by In re C & M Props., L.L.C., 563 F.3d 1156, 1168 (10th Cir.2009).
Perry surmises, but offers no semblance of proof, that a bankruptcy court would treat uncollectible notes worth over $7,000,000 differently than uncollectible notes worth roughly $1,000,000. It is, of course, possible that, in particular circumstances, a bankruptcy court might indulge in such differential treatment. Here, however, the surmised inference is simply not plausible. By definition, an uncollectible note is an uncollectible note (and, thus, worthless). In any event, either of these figures pales in comparison to $134,105,-736 — the sum of the claims of Brown’s creditors at the time of the settlement. In this case, then, the variation in amount is a distinction that makes no difference.4
*13For what it may be worth, Brown himself neither explicitly nor implicitly contradicted his original representation as to the amount due on the Notes. He placed that figure at $902,662 and later sold the Notes for a figure in the same ballpark — $950,000. These events do not support a suggestion that he was attempting to defraud or mislead the bankruptcy court. This is potentially important because judicial estoppel is not meant to be a trap for the unwary and should be employed sparingly when “there is no evidence of intent to manipulate or mislead the courts.” Ryan Operations G.P. v. Santiam-Midwest Lumber Co., 81 F.3d 355, 365 (3d Cir.1996).
To say more on this issue would be to paint the lily. Conjecture, without more, cannot support the application of judicial estoppel. See SBT Holdings, 547 F.3d at 37 & n. 8. Put another way, remote possibilities are not enough. Consequently, we hold that the district court committed legal error (and, thus, abused its discretion) in judicially estopping the appellants, based on a third party’s earlier representation, from attempting to prove that the amount due on the Notes was more than $1,853,282.
In an effort to repair this hole in the fabric of their argument, Perry and CHI urge that the appellants should be judicially estopped based on Yellin’s prior representation. The district court rejected this alternative theory, Perry II, slip op. at 10, and so do we.
The salient facts are as follows. In 1997, Perry, on behalf of CHI, sued the appellants in a Massachusetts state court. He moved for a preliminary injunction to prevent foreclosure on the Fenmore. As part of his opposition, Yellin filed an affidavit indicating, in its background recitals, that the amount due on the Notes was $1,900,000. The state court denied Perry’s motion. Perry did not appeal the denial, and the court subsequently dismissed the case for want of prosecution.
In the court below, Perry and CHI advanced an alternative claim of judicial es-toppel premised on Yellin’s affidavit. The district court did not bite, reasoning that the state court “took no action in reliance on [the $1.9 million] figure” and “ultimately dismissed the case for lack of prosecution.” Id. Perry and CHI challenge this holding. Noting that the state court accepted the filing of Yellin’s affidavit and denied the preliminary injunction, they say that no more was exigible to ground a subsequent claim of judicial estoppel. We do not agree.
Based on the facts recounted above, the first element needed for judicial estoppel is satisfied. The representation about the Notes’ value, as expressed by the appellants in the district court, is inconsistent with the earlier representation that Yellin made in the state court.
Once again, however, the second element is more problematic. There is absolutely no basis for believing that the state court adopted or relied on the $1,900,000 figure contained in Yellin’s affidavit. The absence of any evidence to that effect is fatal. See SBT Holdings, 547 F.3d at 37 & n. 8; Levasseur, 846 F.2d at 794.
Perry and CHI suggest that the state court may have relied on Yellin’s representation in denying the motion for preliminary injunction. But this suggestion is woven entirely out of gossamer strands of speculation and surmise. In addition, the *14suggestion is counterintuitive. Perry and CHI sought the preliminary injunction to prevent foreclosure on the Fenmore, and the amount of the indebtedness had virtually nothing to do with the question raised in that motion. In other words, whether the balance owed was $1,900,000 or $7,000,000 or some figure in between had no apparent bearing on whether the foreclosure should (or should not) be enjoined. Cf. Ross-Simons of Warwick, Inc. v. Baccarat, Inc., 102 F.3d 12, 15 (1st Cir.1996) (listing factors relevant to preliminary injunction inquiry). There is simply no plausible basis for supposing that Yellin’s representation about the amount due factored into the state court’s decisional calculus.
B. Equity of Redemption.
Given our previous conclusion, see supra Part 11(A), the district court’s accounting will have to be reworked. Once the amount due on the Notes is established and subtracted — after all, the Notes were secured by first and second mortgages on the Fenmore and, thus, give rise to a high-priority equitable lien on the foreclosure proceeds — the question becomes how the court should treat the $5,000,000 paid to settle the FDIC’s claims against the Yellin Parties. The appellants argue that this payment extinguished the third and fourth mortgages and, thus, should be offset before calculating the equity of redemption. The district court made no such offset. The appellants assign error to this step in the progression.
The calculation of an equitable accounting is, within broad limits, committed to the district court’s discretion. Tamko Roofing Prods., Inc. v. Ideal Roofing Co., 282 F.3d 23, 39 (1st Cir.2002). Accordingly, “we will not disturb [such a calculation] unless it rests on clearly erroneous findings of fact, incorrect legal standards, or a meaningful error in judgment.” Id.; see In re Blinds to Go Share Purchase Litig., 443 F.3d 1, 8 (1st Cir.2006) (“Because the district court ‘is in a considerably better position to bring the scales into balance than an appellate tribunal,’ we will not normally find an abuse of discretion unless, upon whole-record review, we are convinced that the district court committed a significant error in judgment.” (quoting Rosario-Torres v. Hernandez-Colon, 889 F.2d 314, 323 (1st Cir.1989) (en banc))).
As co-owners of the foreclosed Fenmore, Perry and Yellin are jointly entitled to the equity of redemption. Stripped of rhetorical flourishes, the appellants’ contention is that calculating this figure is a purely mechanical exercise, which requires that payments made to extinguish liens on the property be subtracted before the co-owners can divide the remaining foreclosure surplus. Building on this foundation, the appellants posit that the district court erred in refusing to offset the $5,000,000 payment that they made to the FDIC because that payment extinguished the third and fourth mortgages on the Fenmore.
This argument has a patina of plausibility. Under basic principles of property law, the holder of the equity of redemption does not have a property interest in foreclosure proceeds unless and until all outstanding liens on the property have been extinguished. See First Colonial Bank for Sav. v. Bergeron, 38 Mass.App.Ct. 136, 646 N.E.2d 758, 759 (1995); see also Restatement (Third) of Prop.: Mortgages § 7.4 (1997) (“When the foreclosure sale price exceeds the amount of the mortgage obligation, the surplus is applied to liens and other interests terminated by the foreclosure in order of their priority and the remaining balance, if any, is distributed to the holder of the equity of redemption.”). A junior lienholder has an equitable lien, transferred from the foreclosed *15premises, that attaches to the foreclosure surplus. See New Haven Sav. Bank v. Follins, 431 F.Supp.2d 183, 196 (D.Mass.2006); see also Restatement (Third) of Prop.: Mortgages § 7.4, cmt. a (1997) (explaining that the “surplus stands in the place of the foreclosed real estate, and the liens and interests that previously attached to the real estate now attach to the surplus”).
The appellants take these abecedarian property law principles to mean that because they settled with the FDIC and thereby extinguished the FDIC’s equitable lien on the foreclosure proceeds, the $5,000,000 settlement amount must come off the top before the equity of redemption is calculated. The district court instead treated the $5,000,000 as a part of the distribution of the appellants’ share of the equity of redemption. See Appendix.
The appellants’ construct reflects a kind of tunnel vision. It fails to take into consideration that the challenged calculations are not entries on a closing sheet at a foreclosure but, rather, are calculations made in the context of a judicial accounting. This matters because an accounting is not a rote exercise in arithmetic. To the contrary, it is an equitable remedy, see Braunstein v. McCabe, 571 F.3d 108, 122 (1st Cir.2009) (citing Granfinanciera, S.A. v. Nordberg, 492 U.S. 33, 49 n. 7, 109 S.Ct. 2782, 106 L.Ed.2d 26 (1989)), and equitable remedies “are flexible tools to be applied with the focus on fairness and justice.” Demoulas v. Demoulas, 428 Mass. 555, 703 N.E.2d 1149, 1169 (1998) (citing 1 Dan B. Dobbs, Law of Remedies § 2.1(3), at 63 (2d ed. 1993)).
The district court understood this distinction. It refused to offset the $5,000,000 settlement before calculating the equity of redemption. The court based this determination on a finding that the primary purpose of the settlement was not to extinguish the FDIC’s lien, but to release the Yellins from personal liability on a wide range of loans (many of which were completely unrelated to the Fen-more). See Perry III, slip op. at 1. In effect, Yellin used money that otherwise would have had to be shared with Perry to defray his personal obligations, leaving Perry to settle separately with the FDIC using his own resources. See supra note 1. In the district court’s words, Yellin “ha[d] Blum conduct the Fenmore foreclosure sale and hand over the first $5 million in proceeds to the FDIC” to satisfy Yel-lin’s personal obligations. See Perry III, slip op. at 1. The court found that arrangement to be “the product of precisely the improper relationship and underhanded dealings” that characterized the appellants’ course of conduct. Id.; see also Perry I, slip op. at 13.
This finding is eminently supportable. The only reason why Yellin was able to use foreclosure proceeds to fund his personal settlement was because of his relationship with Blum, who, as a straw for Yellin, held the senior mortgages on the Fenmore.
In performing an equitable accounting, the district court is not a mere scrivener, charged with carrying out a ministerial task. Instead, the court is charged with tempering arithmetic with equity, or, as we phrased it in Rosario-Torres, 889 F.2d at 323, “bringing] the scales into balance.” In this context, we think that the district court acted within the sphere of its discretion in preventing Yellin from unjustly enriching himself, to the detriment of his quondam partner, by what the district court warrantably found were underhanded dealings. See 1 Dobbs, supra, § 4.3(5), at 610 (explaining that an accounting “forces the fiduciary defendant to disgorge gains received from improper use of the plaintiffs property or entitlements”). Accordingly, we hold that the district court *16neither erred nor abused its discretion in refusing to subtract the $5,000,000 payment before calculating the equity of redemption.
C. Joinder.
The Yellins claim that the district court violated the Due Process Clause by joining them as reach-and-apply defendants after the trial had ended. A few additional facts are needed to bring the claim into perspective.
When the FDIC originally filed a claim in the underlying litigation, it named the Yellins, among others, as defendants. But following the settlement of the FDIC’s claims against them, the Yellins were dropped from the suit on June 10, 2002. Two days later, Perry brought Blum, as trustee of Moorings Nominee Trust, back into the case by filing a cross-claim against him. The cross-claim did not specifically name the Yellins, but it did name “John Doe” as a reach-and-apply defendant. “John Doe” was described as “a person or persons, or an entity or entities, to whom funds generated by the foreclosure sale of the Fenmore have been transferred by Blum, for less than full consideration.”
Almost two years later, Perry sought to amend his cross-claim to add the Yellins as reach-and-apply defendants. The district court referred this motion to a magistrate judge. See Fed.R.Civ.P. 72(a). On November 29, 2004, the magistrate judge denied it. Perry moved unsuccessfully for reconsideration but did not appeal that denial to the district judge.
The cross-claim was tried to the court in early 2005. During trial, Perry again moved to join the Yellins as reach-and-apply defendants. The district court suggested that Perry file a written motion to conform the pleadings to the proof. See Fed.R.Civ.P. 15(b). Perry filed such a motion on June 24, 2005. The district court took the entire case (including the motion) under advisement for over three years.
On October 31, 2008, the court issued its written decision. In that rescript, the court granted the motion to join the Yel-lins as reach-and-apply defendants. Perry I, slip op. at 36. The Yellins sought reconsideration, to no avail. Perry II, slip op. at 19.
The district court premised its order on Federal Rule of Civil Procedure 21.5 We review an order joining a party under Rule 21 for abuse of discretion. See Cornelius v. Hogan, 663 F.2d 330, 335 (1st Cir.1981). Within that rubric, embedded legal questions are reviewed de novo. See United States v. Platte, 577 F.3d 387, 391 (1st Cir.2009).
Rule 21 stated that “[p]arties may be dropped or added by order of the court on motion of any party or of its own initiative at any stage of the action and on such terms as are just.” Although the rule permits joinder at any stage of the proceedings, joinder in a particular case must comport with the strictures of due process. Moore v. Knowles, 482 F.2d 1069, 1075 (5th Cir.1973). These strictures include notice and an opportunity to be heard at a meaningful time and in a meaningful manner. Eakins v. Reed, 710 F.2d 184, 186-87 (4th Cir.1983).
For obvious reasons, joinder of a defendant after trial is disfavored. See, e.g., Cabrera v. Mun’y of Bayamon, 622 F.2d 4, 6 (1st Cir.1980). In such a situation, concerns about possible prejudice to *17the late-joined party loom large. 7 Charles A. Wright, Arthur R. Miller & Mary Kay Kane, Federal Practice and Procedure § 1688.1, at 510 (Sd ed. 2001).
The case at bar, however, is highly idiosyncratic. First, the Yellins originally were parties to the action. Second, they were on notice of Perry’s desire to add them as defendants by reason of both his pretrial motion to do so and his mid-trial motion to that effect. Third, in his original cross-claim, Perry named a “John Doe” reach-and-apply defendant, describing “John Doe” in terms that fit the Yellins to a “T.” These circumstances gave the district court an adequate basis for finding that the Yellins had notice sufficient to satisfy the requirements of due process. See Insituform Techs., Inc. v. CAT Contracting, Inc., 385 F.3d 1360, 1375 (Fed.Cir.2004) (rejecting defendant’s due process argument against post-trial joinder because plaintiffs had made pretrial attempts to join him).
Similarly, the record permits a conclusion that the Yellins had a meaningful opportunity to be heard. Blum was a party all along, and the district court support-ably found that he was a stand-in for the Yellins. Perry I, slip op. at 11-12. As such, Blum had substantially the same interests as the Yellins. Blum was their proxy and, as befits a proxy, he and the Yellins shared the same counsel.6
To add to the picture, the Yellins testified extensively at the trial. Consequently, their narrative accounts of the central issues in the case were before the court. That is an important consideration in the due process equation. See Fromson v. Citiplate, Inc., 886 F.2d 1300, 1304 (Fed.Cir.1989) (upholding post-judgment joinder where interests of late-joined defendant and timely-joined defendants were “virtually complete”).
To cinch matters, a party who claims to be aggrieved by a violation of procedural due process must show prejudice. See Amouri v. Holder, 572 F.3d 29, 36 (1st Cir.2009); United States v. Saccoccia, 58 F.3d 754, 770-71 (1st Cir.1995). The Yellins have not identified any evidence which, had they been joined earlier, they could have introduced; nor have they made any other showing of actual prejudice.7
III. CONCLUSION
This is a complicated, hard-fought case. Both sides are represented by highly proficient counsel, but objective appraisals of the facts are to some extent held hostage to the parties’ rancor. The district judge has demonstrated patience and skill in sorting out what really happened and navigating through a legal minefield. We are reluctant to prolong a case that already has lingered for a decade, but there is no other principled course available to us. Perhaps, given the passage of time and the large sums already spent on litigation, the parties have reached a point at which a negotiated resolution of their remaining differences is possible. One can only hope.
We need go no further. For the reasons elucidated above, we conclude that the district court erred as a matter of law in *18invoking the doctrine of judicial estoppel to limit the appellants’ proof as to the amount owed on the Notes. Thus, we reverse that ruling. In turn, this holding requires vacation of the judgment. The district court will have to determine the actual amount due on the Notes at the relevant time and rework the accounting. We take no view as to the amount due on the Notes; although the appellants assert that the amount due is $7,494,435, Perry and CHI fiercely dispute the basis for that figure.
The other rulings appealed from are affirmed, and further proceedings in the district court shall be conducted consistent with this opinion. On remand, the district court may take such further evidence as it deems appropriate.
Affirmed in part, reversed in part, vacated, and remanded. All parties shall bear their own costs.
Appendix
The district court entered a judgment in favor of Perry in the amount of $4,347,126. This amount was calculated as follows:
Combined Proceeds from the Fenmore:
Proceeds from foreclosure sale $ 9,450,000
Net rents collected on the Fenmore $ 1,660,797
Net combined proceeds $11,110,797
Credits, Adjustments, and Amount Due on Notes:
Costs of foreclosure $ 154,440
Amount due on Notes as of foreclosure date $2,262,105
Total credits and adjustments $2,416,545
Equity of Redemption:
Net combined proceeds $11,110,797
Less credits and adjustments -$ 2,416,545
Fenmore’s equity of redemption $ 8,694,252
Perry’s 50% share $ 4,347,126