We had that issue with a checkpilot. They eventually removed him from the program.
On the line side, FOs can no bid a captain in PBS, but reserves can refuse an assignment due to the other crew member (beyond just calling in sick). Captain's can't no bid FOs (because of the order of the PBS runs), but can always tell their CP to either swap out the FO or deal with a last minute sick call.
When I was a pretty new 717 FO, the first month I had a line, I got a call from scheduling telling me they were swapping me to a different (way less desirable) trip and putting that FO on my trip. I asked why but they said that "management" had told them to do so. Turns out that the other FO had been dating two of the three FAs on his trip, without them knowing about each other, and they both refused to fly with him. So instead of recrewing 2 FAs (or 1 FO), they swapped us. Fortunately the captain was cool so it wasn't a totally bad day.