Why Do We Ever Get Interference in a Quantum Eraser? There are many questions on this site about the quantum eraser, but I think mine is not quite answered by any of the other answers on the topic.
Here's the setup:

My understanding of this experiment is the following. If a stream of photons is passed through a double-slit one at a time, but then passed through the BBO crystal, the photon is marked with which-way information due to the nonlinear interaction (basically the act of destroying a photon and making two new ones represents a measurement which collapses the wavefunction). After this the entangled pairs are split; one member of the pair is sent to a detection screen and the other member of the pair is sent into this setup with mirrors and beamsplitters and finally onto some click detectors $D_1, D_2, D_3, D_4$.
Based on this description, it seems obvious that if we look at the photons at the detector screen $D_0$ whose entangled pair photons hit $D_3$ or $D_4$, we'll see no interference pattern: in these cases we have which-way information and there should be no reason for interference.
What confuses me is what $D_0$ registers when you look at the photons whose pair photon hits $D_1$ or $D_2$. I understand that, due to the setup, the two paths have been recombined in the lower portion of the experiment, so that a click in $D_1$ or $D_2$ does not tell you which slit the photon went through anymore. But the upper photons are still marked with this information, so why should we recover an interference pattern? In other words, based on my understanding, the BBO crystal should prevent us from ever seeing an interference pattern.
 A: At the heart of your question is the supposition that:
"basically the act of destroying a photon and making two new ones represents a measurement which collapses the wavefunction"
This is not necessarily the case, if we consider the photon process per Feynman/Dirac, we have excited electron, photon creation and path determination (or vice versa), un-excited original electron and excited receiving electron.  With an intermediary BBO crystal we would add another step with an additional excitation.
The wave function for the above processes may be one or two successive wave functions .... But the fact that a pattern is seen would be evidence for a single wave function at least for the photons on D1/D2.
A: In your 2nd paragraph the photon is NOT marked by its interaction with the BBO crystal, the photons are subject to the EM field which influences the path to take and the path taken is what effectively marks them. In this experiment not only have the photons taken slit A or B but have made a further path selection (due to EM field) as to proceed on a marked path or unmarked path (its 50/50 based on the beam splitter).
An excited atom in a laser is already disturbing the EM field (virtual photons), the creation and transit/path taken by the eventual real photon could be the result of these interactions .... it is well said by Feynman and Dirac that every photon determines its own path within the confines of the EM field.  The EM field is constantly changing due to moving charges in atoms (ex. excited electrons and even somewhat unexcited electrons) and the photon creation is subject to all these forces.  Richard Feynman proposed the path integral explanation, in summary a photon considers all paths (this could be virtual/forces). Most probable paths in his model is based on amplitude summing, considering phase and path length.  One result is that paths that are "resonant" i.e. are integer multiples of the wavelength have higher probabilities.
The upper photons are not marked, we can NOT tell if they came from A or B until afterwards (when D3/D4 click) and this only true for the the ones that have no interference.  What we are seeing is that some photons created in the laser were being affected by seeing (or having a choice) either A or B which results in the pattern ...... OR sometimes they just take a direct path to D3/D4, ie not being affected by the "double" slit at all .... just seeing one slit.  ITS all the fault of the EM field and the quantum (means somewhat random with a pattern) outcomes.
What's interesting about this DS experiment is that adding the crystal and the beam splitters really had a dramatic affect of possible patterns observed. The photons were able to be affected by the 2 slits (pattern) or just only affected by slit (no pattern)!
