Is this a digital multimeter? If so, it has a very high input impedance, and is basically measuring static charges. Measuring voltage with respect to the earth, where the earth connection is a 1" probe stuck in dry ground, is bound to be highly error prone; you'd probably get a more consistent ground by tightly gripping the probe yourself (and not touching the other one's metal). Neither is a very good ground connection, though.
Were you reading 0? (zero) between the ground pin on the plug and the other pins, or ?? (infinite)? The first is a short circuit, the second is an open circuit. I would expect to see an open circuit; a short circuit suggests a ground fault in the trailer—indeed the definition of a ground fault would be a connection between ground and one of the other lines. If you have a hardwired inverter, possibly the transfer relay for the inverter is doing something with bonding ground and neutral and causing a ground "fault"; if it only does that while it is operating, it's not really a problem, but if it does it when you're on shore power, that's not good.