said by jack b:
The only way you could use your Nest or any other low-voltage thermostat to control a room AC is to hook up a suitable load-rated relay inline with the unit's power supply outlet.
You're right, getting such a thing to work well with a modern electronically-controlled A/C unit can be a big pain.
I've run into this before. At work we had an A/C unit replaced at one of our sites where the A/C unit was being controlled with a contactor as part of a more complex HVAC system. The new unit has electronic controls, and would default to "OFF" after power is removed from it.
We actually had to get a firmware modification from the manufacturer of the A/C unit to resolve that problem.