Without researching, I'd say that your "Ground Fault Interupts" are the same as out "Earth Leak Detectors" (our earth is your ground; remember that being "down under" we look down at the sky and up at the ground).
At least GFI isn't to misleading.
Earth or Ground Leak Detector is - it implies that it measures
a or
the leak to ground which it doesn't do (how can it? - it hasn't placed an ammeter at the fault - ie, between your feet and the floor!)
Hence I think the confusion where some think that to be safe, you MUST ground (earth) inverter outputs - or, if you don't, the GFI etc won't work.
The correct tech term for them is RCD = Residual Current Device.
And they work by a current
imbalance - ie, if the two AC lines do not have equal (and opposite) currents.
They use 10mA RCDs for hosptials etc, and usually 30mA RCDs for domestic.
The means that if the currents vary by more than 10ma or 30mA, the RCD trips and opens both lines.
If the currents are not the same, it means it is going somewhere - ie, leaking. And that leak will be to ground/earth.
Hence the name earth leakage detector. That's fine to describe its effect, but bad if interpreted literally.
It reminds me of the old spouse tale that sharks are afraid if dolphins. That comes from the observation that if there are dolphins around, there won't be any sharks.
But the reason is that dolphins are a favourite food of sharks!
The point is that whilst you may be safe (from sharks) with dolphins around, if the dolphins disappear, GET OUT of the water!
But if you see sharks and think that that dolphin will save you - it might be the last spousetale you ever recite.
(Of course in OzLand, if there are no sharks or dolphins around, get out because it means there are crocs! Did anyone see the clip a few weeks back from the Northern Territory where some fishers caught a shark and pulled it onshore, and the beat back a croc as it came up to get the shark? That's why Aussies bring band-aids to the beach, as well as condiments and a grill.)
But back to other shockers.
A GLI should work ok as a "shock" fault detector on an inverter.
It will (should!) not need the 3rd earth/ground pin. (Do your GLIs need a 3rd pin, or do they just have a 3rd pin because wet-areas have grounded plugs?)
The only catch I can see is false triggering due to the non-sinusoid nature of modSquare inverters.
Although RCDs have been around for eons, it took some decades before they became immune enough to noise, surges, spikes etc. Early units were prone to false triggering. They only started appearing in modern homes after the became reliable enough (ie, rarely false triggering).
I have some RCDs like wall socket plug-in timers that I might try on my inverters..... They are cheap (~$20 each) so if they handle the modSquare inverter outputs, any RCD/GLI should.
Remind me.....