The control circuit board in the Norcold is not well grounded and the ground is "backfeeding" from the A/C "run" circuitry. Apply 12 volts to the A/C circuit, no ground, fridge is disabled.
Common problem with dual filament automotive tail/stop lights with
tail/brake assembly on one side missing a ground connection, or an extremely poor ground. Tail lights work normally, turn on the brake lights, the tail lights go off and only the side with ground lights the bulb brake filament.
Tail lights work since the one with no ground has a fairly low resistance ground "backfeed" path, 2 low resistance stop light filaments.
Find a ground point on the fridge control circuit board that goes to 12 volts when the A/C is switched on.
Wall thermostat controlling the A/C?