White smoke usually indicates unburnt fuel, has the consumption increased? The only other thing is water, is the level dropping?

Basically they are both wrong just the issue does not cause a fault code.

This could be linked to the failed turbo. White smoke is water or unburnt fuel, but it's often hard to tell exactly what colour the smoke is, so it might be grey? Grey is oil, that could be linked to the turbo oil seals.

If it's white smoke is due incorectly fuel mix then maybe the air meter system is gunked up or faulty. The MAF on BMW's are know to failing and hard to diag. The easy way to test them is by disconnecting wiring which cause the car to use the default map rather than meter the air. If the car runs better and no white then a new MAF is required.

white smoke due to water would required a test see if that head was leaking, but you would expect to loose some water or see gas in the water if this was the case.

Your cure for the problem is not really a cure these engines will run on the very basic fuel without any smoke you are just masking the problem and its costing you to do it,more than likely the VVT (vanos) on the vehicle is worn or faulty.