Will beat me to it! The inverter you have must put out a modified sine verses a pure sine wave. When you checked the outlets when on shore power and it was Okay that's because the public utilities is pure sine wave. The meter you used did not read true "RMS" voltage. The inverter was actually putting out 120 volts but the meter wasn't able to correctly read it. To read the voltage from a modified sine wave inverter you will need a different meter that will read true "RMS". There isn't anything wrong with you inverter.
Phil
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1983 Newell Coach
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