A third wire is not needed. Simply connect the two AC prongs together and feed. When it's time to light the lamps disconnect the tuner and plug into AC power.
Sounds like an interesting experiment; effectively, an end-fed twisted-pair wire antenna. I'd expect that standard formulas for element length wouldn't apply, as there would be some slight capacitance between the two wires... I doubt the inductance imparted by the twist would be more than miniscule, though. Then again, it might be slightly more broadbanded than a single wire of similar electrical length.
Here's a thought - make one up using neon bulbs in place of the incandescent ones; perhaps with practice, you could determine the state of tune by the light patterns on the antenna!