Apologies. Apparently my opponent turned on vacation mode after filing their most recent move, which is probably why the game is behaving the way it is: I win, but as far as the game knows the only other player is an AI, so there's no reason to hand off control to the other player.

Follow-up question: I'll assume that my opponent will eventually turn off vacation mode and allow the game to conclude, but if not, how do I clear this slot from my list of games? (For what should be obvious reasons, I'd prefer not to

The same for me. After last opponent surrendered and got AI control - I had space ship launch in few turns and not sure how to claim the victory. Going to just surrender.

Is there actually a record of victory on GMR? I suppose otherwise there's no record except in our own minds, but I'm irritated by my opponent for going on "Vacation" just so they can avoid losing in person. I suppose it shouldn't bother me that I need to surrender to free up the game slot, but...it does.

Even if your opponent had not turned on Vacation mode, a similar situation would occur, as described in your 1st post. It happened to me several times. What I did in those cases is, send a message to my opponent and a screenshot of victory (if I need to convince him/her OR if this was a League game).

As for keeping a record of the winners & losers on regular GMR games, I am not aware of anything except for personal knowledge (remembering the games, keeping notes about them etc.) However, we do have an unofficial GMR League, that players organize their own games, but limited by League rules / themes and the win / loss record is kept online:

https://civilizationleague.org/