That should create an interference pattern. If, however, the other stream with the entangled pairs is measured in such a way as to "preserve the momentum information," it's akin to observing which slit the initial photons went through and destroys the interference. Birgit Dopfer did this in 1998.
A wrinkle in the experiment is that she useda coincidence detector (basically an AND gate) to filter out all the non-entangled photons. A guy, Dr. John Cramer, was working on developing a system that didn't need that coincidence detector, and what he found was that there is a kind of anti-signal which fills in the spaces in between the interference fringes and masks out the signal. I'm quite confused by this: how does it exactly mask it out, and if the coincidence detector filters the anti-signal out, how is it that the anti-signal responds to the measurement on the other stream? My conclusion was that I just had to build the damn thing and find out.
It seems to me like there should be some way around this problem, and I wonder whether there's some sort of pattern between entangled photons vs. non-entangled, anti-signal photons. Maybe there's a timing relationship? It seems highly likely that Cramer explored this fully, but I just can't let it go. Maybe it would be possible to train a neural network to detect some sort of difference that us humans don't otherwise see.