..and that is basically the same advice I gave when I first heard about the problem - but I've been able to reproduce it now and its turning out to be pretty strange. One of those bugs that takes a decade or so to happen
I think it could be related to failing NVRam batteries, and basing video register choices on the system video fields. Or perhaps certain monitors getting annoyed with the video profiles baked into Apex.
In any case I didn't see this happen back in the '90s but now it seems common, and happens on a cheap LCD TV tested here. I notice that sometimes it can happen a second or so after the video mode is set, which suggests the signal is slightly out of limits. I've seen it happen in both 50 and 60hz modes, but most often happens when flipping between them.
I snapped some new video profiles using ScreensPain and replaced the ones in Apex with those, and avoided the resync step which was needed when applying the old profiles (probably because of the order the regs were being set originally) - and it seems to work now on my TV.
It would take me a bit more digging to find out exactly which bit was going wrong (the profiles themselves or register loading & related inputs) but since it works now I'll probably just go with that, once I've had some positive testing feedback.