When the Joker kidnapped you in my time he took the opportunity to stick some stolen Cadmus tech into you. Small enough that nobody who didn't already know it was there would be able to spot it.

[Terry doesn't like having to tell Tim this. He'd already been through a lot. Telling him that he wasn't safe even inside his own head was even worse. But it had to be done.]

A chip that eventually turned you into an exact copy of him.