With a GFCI connect between the source and the point where the ground and neutral are tied together, this will just about always result in the GFCI tripping. This is because the return current which is normally confined to ONLY the neutral is shared between the ground and neutral which the GFCI sees as an imbalance between hot and neutral. A very common thing to happen is that a service panel is updated and AFCI (arc fault), GFCI, or a combination of both breakers are installed. An AFCI breaker has some GFCI protection but it does not detect as low of an imbalance as a true GFCI breaker does. Anyway, in the above scenario we will have those breakers tripping for no apparent reason. None of the old breakers tripped. The problem is often in an outlet box where the bare ground wire has shorted to the neutral terminal. This was not a problem before AFCI and GFCI breakers were installed. It was likely this way since the original install and was not a problem.Naspook said, "If the connection is made at the distant device a GFCI device will trip due the unsafe condition if you are lucky."
As others explained, that could happen if part of the return current went through ground rather than neutral.
But the opposite is scarier: If an unlucky person touched a life wire at the remote site, current would flow through them to ground (or earth if you like). But as ground and neutral are connected there, the harmful current may return through the extra earth-neutral connection and through the neutral wire. After all, a solid copper neutral wire likely offers a path of less resistance than traveling through the soil. And the remote grounding rod might have a better connection. That means that the GFCI will detect matching opposite currents on life vs. neutral, and fail to trip.
Does this scenario make sense? If yes, then I'd be safer disconnecting the remote neutral-ground short.