Another possibility is that our setup has been made especially sensitive to this by the PBS after the modulator. The RAM typically gets made by unwanted polarization rotation. If we have tuned a waveplate to split the power after the EOM 50/50, it means that we are also in the configuration where the polarization -> AM conversion is maximized. We can reduce this by trying to do a polarization insensitive split or by having separate EOMs in the two paths.
This is almost certainly what is happening---I guess I assumed that was the case. We can trim the offset that develops away by making slight changes to the HWP before the EOM, counteracting the rotation that happens in the modulator. I have seen lots of setups with PBSs after modulators, so I didn't foresee it being a problem when we designed it in the first place.
I guess we could use a power splitter instead of the PBS, then put a waveplate directly after it in the AOM path (before the second PBS that directs the beam to the AOM). Using the waveplate, we could make small changes to the power that gets sent into the AOM (vs straight through the 2nd PBS into a beam dump) to finely balance the power without having a maximal linear coupling from the polarization rotation in the EOM.