I've seen something like this with "company radios". The current draw from starting the vehicle causes the radio to turn on even if "ignition sense" is not wired. Could be a voltage drop causing it, try doing other current drawing things, headlights, vehicle start etc. Dunno.
It goes off during crank, even though the starter is a stomp pedal directly on starter.key is on/off. But i will try starter tomorrow with key off to see what happens. Stereo has a battery connection and an acc connection.