I’m a newbie, so get out your grain of salt, but here’s what I think:

If both sides are on “the” Internet, then either can read the other’s blockchain. And therefore either could reject a spend that has already happened on the other. If BTC1 knows that BTC2 is doing so, it would be foolish for BTC1 to not do the same.

It would also be possible (with software changes) for a spend made in one chain to automatically remove those coins from existence in the other.

HOWEVER, since no government is involved in maintaining the chain, the only way such a fork could happen would be for one of the sides to block any packets related to BTC, right? And with today’s encryption methods, VPNs, and things like TOR, the only practical way of doing that would be to block ALL Internet packets.

So, if you live on side two, the only way to spend on side one would be to travel to there (in this scenario, probably a dangerous thing to do).