Since 2x has decided not to implement replay protection, transactions will be indeed compatible to both chains. If a transactions is simply broadcast to the other chain, it will enter the mempools there as well and may eventually get confirmed. This doesn't mean that one is completely helpless though. Here are some methods of making transactions valid only to one chain:

1. Input derived from a coinbase transaction
   Coinbase transactions will only ever be valid on one chain. Therefore, including an output that derives from a coinbase transaction will make a transaction only valid on a single chain. Coinbase transactions need to mature to 100 confirmations before their outputs may be spent, so transactions based dependent on a coinbase will become valid about 17 hours later at the earliest.
2. nLocktime and doublespend
   If one of the blockchains has significantly more frequent blocks than the other, one can create two transactions sending to oneself: One with a nLocktime for the next block on the faster chain, and another that spends at least one of the inputs from the first transaction with a very high fee. First, you broadcast the transaction with the nLocktime which will hopefully get included in the next block on the fast chain. Once it is confirmed, the doublespend is broadcast with a very high fee to the slower chain, hoping that it will get confirmed before the height of the nLocktime of the first transaction is reached. After waiting some more confirmations for safety, the outputs of these two transactions can now be used to create transactions specific to only one chain.
3. Doublespending
   Create a pair of transactions with different recipient addresses that you own that spend the same funds. Broadcast both at the same time and hope that one gets confirmed on one chain while the other gets confirmed on the second. If unsuccessful, repeat.

Generally, as soon as one has some unspent transaction outputs that are unique to one of the chains, these can be used to make other transactions safe as well.