New Zealand's new approach to the proviso has been revealed by the Privy Council. The Board fearlessly disclosed parts of the Supreme Court's decision in Matenga v R  NZSC 18 which has been embargoed pending a retrial. The new approach is set out for us in R v Barlow  UKPC 30 (8 July 2009).
The Supreme Court of New Zealand has adopted, with some modifications, the approach to the proviso taken by the High Court of Australia in Weiss v R (2005) 224 CLR 300 (noted here 16 January 2006). For reaction to Weiss in Australia, see “The Problematic Proviso: the vice of Weiss” (2007) 140 (Autumn) Victorian Bar News 32 by Phillip Priest QC, who comments “Weiss caused a shockwave to sweep through the ranks of criminal appellate lawyers and judges. The High Court radically altered the ambit of the proviso, sweeping away decades of accepted wisdom.”
Two departures from the Weiss approach are established in Matenga (according to Barlow).
The first departure from Weiss
This has several aspects. Not all errors at trial qualify to be called miscarriages of justice. Errors that "plainly" could not have affected the result of the trial don't count as miscarriages in the relevant sense. "A miscarriage is more than an inconsequential or immaterial mistake or irregularity." (para 30 Matenga, quoted in Barlow at 19)
If the error counts as a miscarriage in the required sense, namely that it was capable of affecting the outcome of the trial, the next question is whether it actually did have that effect "in reality": (Matenga 31)
The second departure from Weiss
This is quoted in para 20 of Barlow, from para 33 of Matenga:
What about R v Bain?
In R v Bain (New Zealand)  UKPC 33 (noted here 11 May 2007) the Board declined to apply the proviso, emphasising (para 25) that the appellate court "is not the ultimate arbiter of guilt, save in the practical sense that this is the effect of applying the proviso, or ruling that the new evidence could not reasonably have affected the result."
In Barlow (21) the Board was coy about the whether the Matenga approach was different, but it thought the result in Bain would have been the same under Matenga:
The Board showed no sign of eagerness to apply Matenga in all the legal systems it serves.
The trial fairness requirement
Upon identification of qualifying error or errors, the application of the proviso is addressed by first asking whether the trial had been made unfair (Barlow 58). Fairness is judged in the light of the proceedings as a whole. The test is whether the departure was such as to go to the root of the proceedings, in the sense used in Wilde v The Queen (1988) 164 CLR 365, 372, per Brennan, Dawson and Toohey JJ and approved in Howse v The Queen  UKPC 31,  1 NZLR 433 (PC).
If the court is satisfied that there was no such unfairness, it proceeds to the next question, whether the error may have actually affected the result of the trial.
A potential inconsistency with the new role of judicial fact finding is the inquiry into fairness, which necessarily requires determining whether the miscarriage could have caused the jury to misapply the law or to reason wrongly or to give improper weight to items of evidence. This goes back to the former approach of asking how the jury, not the judges, would have decided the issue.
Appellate judges as jurors
So, on appeal the judges do not try to guess what a jury would have decided. They have to assess the whole of the admissible evidence and decide whether they are satisfied beyond reasonable doubt that the conviction is correct.
Barlow illustrates the way the Board embarks on this task. It used the judge's summing up to the jury as a convenient summary (this was a fresh evidence appeal, not a case where the summing up was alleged to be defective), plus obvious circumstances that the defence relied on. The approach was not a minute and detailed examination (in the judgment) of the arguments for and against every possible inference (that was the approach that the Court of Appeal had wrongly taken in Bain), but instead the Board noted four "very obvious aspects" of the prosecution case with which it was "particularly struck" (67 – 71).
The new approach to the proviso will expose appellate judges to criticism by people who disagree with their evaluation of the evidence. It can be seen as a qualification on the right to trial by jury: you have the right to jury trial, but if that trial is not conducted according to law in a material way, your guilt or innocence will be determined by judges on appeal. There need be no second attempt to give you a lawful trial.
So when will retrials be ordered? Given that they are now fact finders, if the appellate judges have a reasonable doubt about the appellant's guilt, why don't they order an acquittal? Ordering a retrial looks like double jeopardy. In Barlow that issue did not arise.