I recently looked at putting a GFCI outlet on the output of an inverter for some level of safety running Christmas lights on the boat. The inverters I looked at were the smaller hardware store versions (400 watts and 150 watts) but they were not consistent in how the grounds were wired. One inverter had the DC ground connected to the AC green wire. The second inverter seemed to have no connection at all between the DC ground and the the AC green wire. The DC ground or the AC green may have gone to the inverter chassis somehow but I couldnt tell since most of the chassis was plastic.
Also, the AC output (hot and neutral) were floating with respect to the green wire on both of these inverters. Ie, the AC output had no low impedance reference to Green. Note that either at your house or on the boat, the AC source (like the marina breaker) will have the green and white wire connected and this is what references the AC current carrying wires to ground. GFCI has to have the reference in order to work.
So if I had connected a GFCI ACsocket to the output of the inverter by directly wiring up the black/white/green wires, it would likely have never tripped under any condition since the hot wires had no reference to the boats ground, green earth wire or water. Not really related to this case but to make the GFCI trip for my Christmas light case, I would have had to tie the GFCI white wire to the boats ground (which was connected to the water through the outboard).
For the OP, as suggested already.. Your toaster likely just has a two wire plug. Take an ohm meter and measure between each of those two wires to the toaster chassis. You should see an open circuit for both wires. Any resistance at all - even very high resistance - something is wrong.. time for a new toaster.
Second, connect up the AC source/circuit that does not cause the shock. Put the meter on AC volts and measure between black to white (hot to neutral), black to green (hot to earth ground) and then white to green (neutral to earth).
If this was at your land based house, you would see about 110 AC black to white, 110 AC black to green and close to zero white to green. Your AC circuit that does not shock may look like this. The inverter output may look more like 55 volts black to green and 55 volts white to green. There really isnt anything wrong with the way the inverter works.. but that may be related to why you get the non dangerous tingle shock???