Wow, that is weird! Sounds like you have an intermittent wire problem, perhaps a short somewhere or a loose/damaged pin.
From the FSM, Rear View Mirrors section, Pinpoint Test F, disconnect the connector (C516 left, CC622 Right) and turn the ignition ON. Turn on Rear Defog. Check voltage at Pin 8 (brown/white). If less than 10V, check BCM fuse 59 (15A). If that fuse is OK, there is a problem in the wiring between the fuse and the door mirror.
With the ignition OFF, the resistance between Pin 2 (Black/Violet) (body side of the harness) and ground should be less than 5 ohms. This verifies a proper connection to ground.
Check the "jumper harness" (I'm guessing this means the wires between C516 and the connections on the back of the mirror glass) for open circuits, loose or damaged pins. If no wiring faults can be found, replace the mirror glass.
That's about it as far as the factory manual is concerned. . .
EDIT: If there's a short, you should get a reading of a few volts or less. If you get a reading of -12V, the probes on the voltmeter are likely backwards. I can think of no way you could get a negative voltage coming out of the fuse when the other mirror is working correctly and both are being fed by the same circuit. Of course, be sure you have a good clean ground point to connect the negative probe.
It sounds to me like there's a pinch/short in the wiring that occurs when the door is opened. Or, maybe a loose or damaged pin in one of the connectors.