The point is that if you sign the transaction in one fork, the recipient can submit the same transaction to the main chain and also get your coin there.

The only way around it is to have a different signing system on the fork.  For example, the rule could be that a script is valid on the fork if the signature is valid for the script or the script with "ALT-CHAIN-1" appended to it.

Transactions on the main chain would continue to be valid on the fork, but once you sign a coin to a script with the suffix, then they are no longer transferred in lock step.