More speculation regarding the three wire toaster plug since no measurements..
Assuming your toaster is fine, another possibility that I would be surprised with.. but one of the two inverters I had would have done this..
On your inverter, un plug it and take an ohm meter and measure between the AC output earth green and the DC 12 volt ground into the inverter. If this connection is an open (which was the case on one inverter I have but not the case on the second inverter), the green wire on the inverter output is not connected to your boats ground like it should be.
In this scenario (which we dont know if it is the case.. more speculation), the floating green out of the inverter capacitively picks up the AC signal and if the toaster is wired correctly, the floating green wire that has the coupled AC is connected to the chassis of the toaster.
This is not really dangerous (except that it would be for a real fault) but the capacitive coupled AC on the toaster chassis would result in a mild shock if you touched it and anything else related to the boats ground.