A GFCI outlet should trip if there is a voltage difference between the
ground wire and neutral. Any flow there indicates a potential problem
and should cause it to trip off. Without a ground connected, this
effectively prevents the GFCI from working properly.

I believe that your description of how a GFCI works, is incorrect.

Everything I have read about them, says that they trip when they sense
a *current* imbalance, between "hot" and "neutral". They do not sense
voltage differences between ground and neutral. They can operate
without any ground connection at all.

Basically, they run these two wires through an inductive transformer
core in a "balanced" arrangement, and have a secondary sense wire in
the same core. If the current flow in hot and neutral is the same,
the fields around the two conductors are equal but opposite in phase,
they cancel out, and no current is induced in the sense wire. If
there's any hot-to-ground current flow "downstream" of the sensing
transformer, then the current flow in the neutral wire is reduced, the
two currents don't cancel out entirely, a current is induced into the
sense wire, and the GFCI detects this current and trips.

Citation: http://www.cpsc.gov//PageFiles/118853/099.pdf from the
U.S. Consumer Products Safety Commission. This fact sheet
describes the current sensing design, and specifically points out that
a GFCI can be used to protect a two-wire circuit which has no ground
connection at all. All of the other references I have seen, agree
with what the CPSC is saying here.

[An episode of "CSI", years ago, got this wrong... the lead story was
a murder which was arranged by tampering with a drill that was
plugged into a GFCI - cutting off the ground prong on the plug and
then arranging for a current leak. In fact, a GFCI would have
tripped under these circumstances, even with an un-grounded tool case
and even if the ground connection on the GFCI itself had been cut.]