What he shows in that paper is that the order of the measurments is irrelevant [1]. So he does solve the problem in the reverse order where it can be done easily writing a few quantum states [2].
Note that QM is not about causal connections, is about correlation. Once a pair is correlated, the correlation may appear when measurements are done. But it's not that observing one outcome here and now "causes" a particular outcome there and later (or before). One doesn't need to keep superpositions to think it "correlation" terms (instead of "causal" terms).
[1] Using the projection postulate in the derivation: "Say we have indeed measured on B and got OB = bJ . The state then collapses onto ..." He concludes: "None of this looks very surprising, but we want to stress that the total probability to find OA = aI and OB = bJ does not depend on the place or time at which the measurements occur."
[2] He also uses the projection postulate here in the usual way: "So the experimental outcome (encoded in the combined measurement outcomes) is bound to be the same even if we would measure the idler photon earlier, i.e. before the signal photon by shortening the optical path length of the downwards configuration. Then, if the idler is detected at D4 for example, the above state ‘collapses’ onto ..."
And yes, you don't need to ask "when does the wavefunction collapse" to manipulate joint probabilities of measurements that happen at disparate locations and times. In fact, that's my objection: If you do ask, you find that there is no consistent answer! (And I suggest it's because wavefunction collapse is not a thing the universe does).
Re: "interference tagging"-- do you have a link to some material? (I'd love to understand something specific before commenting).
EDIT: Also, I didn't see Appendix B at first-- The authors do understand and even advocate the Everettian view! Though I still don't quite understand/agree with their earlier timidity about finding and interpreting conflicts with Copenhagen.
I'm not sure what's the problem. Why do have to "ask" if the answer doesn't really matter? What answer more consistent than "it doesn't really matter" would you like? Anyway (standard) QM is a non-relativistic theory, QFT may be more satisfactory from that point of view.
Re: "interference tagging" - what I mean is that first you detect the photons and later check if they "did happen" to go through two slits (interference appears) or one slit (no interference). But the interference pattern is not visible for a single photon and at that point the individual events are still a superposition of both possibilities (so for the events at a certain position part of them will be in the end identified as coming trough one slit and some of them from both). Only after the second measurement is done you know how to group the previoulsy recorded events to see the interference. It's not that the later measurement causes interference to appear. Or at least it doesn't affect at all where the photons were detected, it just lets you know how to group the existing events to make it apparent (selecting only those where, once the full mesurement on the pair has been done, the path taken remains uncertain).
Edit: maybe this picture helps https://upload.wikimedia.org/wikipedia/commons/thumb/c/c8/De...
If all the events are taken together there is no interference pattern. But when they are grouped according to where the second photon is detected in two cases there is still no interference but in the other cases complementary interference patterns appear.
You tell me; Copenhagen is the one that says collapse exists. It sounds like maybe we are on the same page that collapse isn't necessary to explain quantum mechanical observations? In that case, we are both Everettians :).
In some cases it is a safe approximation to ignore those extra states for the remainder of our experiment/calculations, but with a small change to the experiment we can make that a bad approximation.
a) You do the measurement first on the "screen" side (and project the quantum state of the pair of photons according to the measurement, the "extra universes" disappear). You do then the measurements on the "idler" side (and project again the quantum state according to standard QM).
b) You modify slighly the setup to reverse the order of the measurements. You do the measurement first on the "idler" side (and project the quantum state of the pair of photons according to the measurement, the "extra universes" disappear). You do then the measurements on the "screen" side (and project again the quantum state according to standard QM).
QM predicts that the outcomes in the original experiment (a) and the "reversed" experiment (b) are the same. And those predictions are verified empirically.
Before measurement, the state of the particle system is Σ α_i|i>. Copenhagen says, "after you measure the system, all but one of the α_i go to zero". The authors don't do that, and in fact say that you can't. Unless I am misunderstanding something, they keep all the α_i around at all times and project the measurement for each particle separately, regardless of whether it has been (or will be) measured somewhere else. There is nothing philosophically or physically wrong with this, as they point out, but it is different than what Copenhagen says you should do when you measure something.
And if you look at what it means (which they refuse to do), you'll see that after Alice makes her measurement, Bob's quantum state is still explainable in terms of a superposition. When is the measurement (joint or individual) "finished"? Answer, per the authors: Never. The superposition permeates spacetime. That's how we escape the need for a causal connection between Alice and Bob's measurements, and, naturally, that's how many worlds does it.
(And if we take it further and ask, when Bob makes his measurement, "What is the state of Alice's particles?" we'll see that she is in a superposition of being entangled with each of the superpositions of the particle, which remains in superposition before, throughout, and after our ~measurement of~ entanglement with it).