On chain A blocks bigger then 1mb will be valid
On chain B blocks bigger then 1mb will be invalid
Now when you sign a transaction and broadcast it, it will get executed on both chains.Unless the transaction has coins in one of its inputs that already went through a block bigger then 1mb. Then the transaction is only valid in chain A. This way you can double the value of a coins that are spendable on both chains. Because now you can sell them once on chain A and once on chain B.