Paradox/inconsistency when embedding QFT in Newtonian gravity

This is not a question per se — I am trying to recall an example from a lecture long ago about the inconsistencies that arise when you try to use QFT with Newtonian gravity.

It involves an electron-positron pair falling side-by-side in a uniform gravitational field, and from QED we have the $e^+e^- rightarrow 2gamma$ annihilation.

This is supposed to be where the inconsistency arises – but while I have a rough sense what is wrong, I can’t remember the precise framing. It has either something to do with (a) the redshift (or not) of the two photons, and energy conservation (b) the massless photons being entirely unaffected by Newtonian gravity and don’t fall

I guess at this point I’m too used to thinking in terms of a GR framework that I can’t put my prejudices aside to see the problem with naive eyes…