Found The Phantom Start Gremlin

I put a voltmeter on the coil and switched her from 'off' to #1 battery - nothing. I turned the key to light her up and she was running before I could release the key. This would have explained why I never heard the starter - it could have hit just for a fraction of a second.

She was pumping cooling water as she ran, so I goosed her momentarily to energize the alternator. The needle jumped right up to just a little under 14v. So, evidently no damage by her coming alive on her own and me switching the battery off while she was running.

I let her warm up a little. While I was peering into the engine compartment, looking for clues the starter engaged on it's own, trying to start the running engine. I disengaged the starter by switching the batteries to 'off' but the motor kept running. When I turned the key off, she went silent. I lit her up again and got the same results.

I went up and examined the keyless ignition switch on the flybridge. It was very warm to the touch. Suspecting some of the recent heavy rain got into the works, I removed the switch and took it to the bench to check for conductivity; the ohm meter showed resistance, telling me the circuits were not completely open. Apparently with the ability to pass a slight current, the sizzling and cooking going on at the contacts would occasionally incite full juice to pass.

With the flybridge switch disconnected, I again lit her up and let her run for a spell. She appears to be happy now, pumping cooling water, charging the batteries and not trying to start on her own. When I rewire the new flybridge switch, I'll put in a separate starter button - to keep the ignition and starting circuits isolated.

The gremlin:

I decided to cut the switch open to verify that it was the culprit.

It's pretty safe to say water in the switch was the cause. There is a drain hole in the switch, but the switch was mounted so the hole was at the top. The B+ wire should be at the top.