Is Post's correspondence problem recognizable?
We can recognize acceptable inputs of PCP by exhaustively checking all valid possibilities (preferably by non-deterministic TM). That is it!. If the input is acceptable then the TM will stop, if it is not then the TM may not stop. Thus PCP is an RE language. Recognizability should not be confused with decidability.

We can do the same with deterministic TM by essentially simulating each thread of NDTM one step at a time.