Yes, if it's an "electronic transformer" with an electrical path between input and output, then a ground fault could cause the GFI to trip, so long as it is a DC-sensing type. (Note that many will detect pulsing DC, such as you would get as the raw output from rectified AC, but they won't trip on pureDC).
Such power units aren't really transformers in the true sense of the word. With a real, genuine transformer with no direct electrical connection between the windings, a ground fault on the secondary will not trip a GFI on the primary.
Sparky,
You could even set up one of the SWER systems we were discussing some months back and have the secondary of the xfmr feed into the single wire. You could still put a GFI on the primary and it wouldn't trip, even though the full load current is flowing through the earth on the secondary side.
