If your trem is giving you trouble, there are a number of things that can be done to try make it better behaved.
Sometimes, however, the issue is hiding a little more deeply that others.
The tremolo (let's not get into the vibrato/tremolo argument—we all know it should be vibrato but the nomenclature and popular opinion has won the day) on this guitar wasn't the most reliable. It's a copy, based on the vintage Strat-style bridge.
After a little investigation, the main culprit seemed to be some 'swarf'—a rough curl of wood left from the machining process—had been nicely lacquered over and was actually rubbing against the large trem-block.
A little clean-up with some sandpaper and the bulk of the tuning problems were sorted. Nice.