Mmmm.. Gotta think about this.

A GFCI simply checks for equal current on the two main prongs (hot and neutral). If they ever differ, it trips.

So what happens with a power injector, which is sending current one-way down a coax, and using earth as the return path?

Sounds like a potential current imbalance at the outlet, though I suppose that would depend upon exactly how the injector is converting the mains-AC to DC-for-the-injector.

If it uses an old-school transformer, no problem. If it used something akin to a diode and lots of serial resistance (or waveform chopping), then perhaps that would result in a one-way flow from hot to earth (and trip the GFCI) ?