This was also my first idea, but it cannot simply be partial reflection, given that the frequencies of idlers and signals are different, and that there is a filter before the s1+s2 detector, filtering out any idler contribution. That's why I suspect a frequency change i2 -> s1 which is not excluded, because for that you need a non-linear interaction, and NL2 IS a nonlinear material. Moreover, the interference only appears when i2 is EXACTLY aligned in such a way, that the momentum and energy relations in the NL2 xtal are satisfied to allow for such a i2 -> s1 transition. But, as I said, I don't know enough of the detailled physics of these xtals to work out exactly what is going on.
BTW, I also agree fully with you that the author makes quite some errors by considering the s1 and s2 photon states as "independently generated" by a "spontaneous process" : in that case, in no way ever there could be any interference. They are produced in superposition (by the pump photon), but, as you also point out, with random phase relation due to different points of conversion.