Thanks, all. Fast responses!

The power is coming off the alternator, and before the split in the wire there is 12V. There is no, I mean zero, power at either end of the hot lead (after the split) when either side is hooked up to a choke. There is just under 12V at both ends when neither end is hooked up.

Denny, I'll scrap the whole thing and try it your way. I'm still curious as to why I'm not getting any power when one or the other is hooked up, though; it kinda makes me uneasy.