What happens to my bitcoins if there's a permanent fork in the chain? Let's say the year is 2034 (optimistic?) and the New New World has adopted Bitcoin as its currency, and now there's a civil war, and naturally the two sides want to have their own version of Bitcoin ... hence a fork.

A permanent fork as far as anyone can tell.

What happens to my bitcoins in such an event?

If originally I owned BTC 1,000, do I now own BTC 1,000 and BTC2 1,000 separately after the fork, or is it just BTC 1,000, or do I have to choose between the two chains (i.e. I get either BTC 1,000 or BTC2 1,000 but not both), or do I lose all my BTC unless I redeem them for either BTC1 or BTC2 ... or what?

I understand that forking Bitcoin into two or more currencies might reduce its value more than proportionately against real goods and services, which is to say that BTC2 10,000 might not buy you what you could have got for BTC 1,000 before the fork event, and we could get into the economics of that, but that's not my question. My question is specifically about the nominal and not the real, i.e. how many bitcoins I now own as opposed to what my bitcoins can now buy.
The question is too speculative to answer. The groups could institute whatever rules they want.

If you assume there's a group that's willing to modify the software to create a permanent fork, presumably they are doing so for some reason. And whatever that reason is, that will affect what changes they make that cause the fork.

They could decide to allow people to transact on both chains. Or they could demand that you send the Bitcoins to a Bitcoin eater in the other chain before they allowed you to move them.