[jabbany]

Unfortunately not in the case of my furnace...

It was upgraded with a rather new controller board at some point despite the machine being rather old, and is very unhappy about any extra loads on the heating control wire. In fact the controller supports all the things (even potentially controlling AC) but there are only 2 thermostat wires in the wall...

[doubleg72]

So why not run a common off a plug in transformer?

[jabbany]

The old thermostat is placed quite far from any place I could plug in a transformer to... (otherwise, yes, that would have been the solution)

My current setup has the relay right next to the furnace, where there's plenty of mains lines to tap into for continuous power and this is fine since it doesn't measure anything and just gets commands from hass.