Today while trouble shooting, I was able to get a start after 8 hours on the charger, (while the charger was still hooked up). I've had the tachometer unplugged while checking fuses. So I turned the engine off to hook up the tachometer again. Everything dead again. I think I need a battery.
Anyway when I plugged the tach connector in, the back light came on, (this is with the key OFF), then the needle did a very slow sweep to 8000 RPM, then kicked back to zero and the back light went out. The whole sweep maybe took about 15 seconds. Not worried about that knowing there is a direct connection to the battery, besides the switched wire.
I was concerned at first that the tach would draw a bit of power even when key off. I suppose there is a relay inside that would take care of that, though.
Still, it was strange.