Wiring HVAC for smart thermostat
Alright - this installed and working.

It took a little bit of time to figure out what was going on with that crazy wiring without the benefit of a wiring diagram. We reverse engineered the wires based on where they were on the old thermostat.

The thermostat came with a c-wire adapter which basically allowed be to wire the unit using the existing 4-wire cord. I'm pulling the 24v from the transformer and, so far it's not running under-voltage. If that changes then I guess we'll have to install another transformer. So far so good though. Thanks for the comments!