If you watch the rotary spinning under fluorescent light, you'll be able to
see if it's truly running in sync (just the rotary alone, no Tesla coil firing).
The light pattern will rotate if you lower the rotary voltage so it drops out
of sync, and you can hear the hunting sound when it's almost syncing
as you raise the voltage. (I'm referring to the motor voltage not the
phase shifter variac setting). You can just unplug the motor and you'll
see the light pattern start rotating, then freeze when you plug the
motor back in and it syncs. Some people have a very very very hard
time seeing these patterns under fluorescent light however.
(Maybe you did all these verifications tests already).