The creation of a "double blockchain" is a problematic situation that Bitcoin has been trying to prevent for a long time. Not only it creates confusion amongst investors and casual users, but it also opens possibilities for replay attacks on both blockchains.
If a user signs and broadcasts a signed transaction in one of the blockchain, that transaction can be replayed on the other chain without the consent of the private key holder, since the cryptographic signature for said transaction his already public on the other blockchain.