I have never had a marine refrigerator, but I know something about house refrigerators...so here is a theory:
Surfaces on the outside (or underneath) the refrigerator can be cool enough to be below the dew point, and therefore coalesce moisture out of the air into liquid water. Designs can include dew prevention heaters...low wattage heater strips that prevent the outside surfaces from getting as cool. If this marine refrigerator has dew prevention heaters, it would make sense that they would be activated on shore power, but not be activated on battery to save power. So dew accumulation would only happen on battery power. Just my theory. Perhaps the refrigerator operating manual would say something about this, or list a different wattage requirement on shore power vs battery.
Ray
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"Knot Easy" 2000 Horizon 240 Volvo 5.7GS /SX
tow: 2017 Honda PILOT EXL-AWD
prev. boats:
'87 Chaparral 198CXL 4.3 OMC Cobra
'69 Jetstar 16ft Ski Boat, 115hp Yamaha
'68 Aluminum Jon Boat, 3hp Sears
'64 Water Wings