Bidirectional half-duplex communication between PIC16 and an FPGA over a single copper track
As long as both the PIC and FPGA support tri-state buffers (i.e. can make them Hi-z) which is highly likely, then yes, you can easily do half-duplex with a single wire.
The safest way is to use open-drain outputs whereby each device only drives a logic 0 or releases the pin to hi-z, that way you can never have bus contention. A resistor is then used to pull high. If one or both devices have built-in pull-up resistors you can enable, that would save having to bodge a resistor onto the board.
If you don't have usable pull-ups, you would have to have some arbitration (e.g. master-slave) to make sure that both devices don't talk at the same time. When a device isn't talking it sets its pin as Hi-z, while the other drives high and low.
There are examples of fully formed single wire comms, such as 1-Wire which can perform bidirectional comms at 16kb/s.