The sort of transformer that we are discussing has 4 coils placed on a single core. Two of these coils are rated 120V, and two of these coils are rated 16V.
As shown in the diagram in post 29, with the transformer connected as in the original post, the 'center tap' of the transformer will be at 60V relative to the grounded neutral of the wye source.
In theory, if you have a 120/240V single phase load, it will operate just fine between the center tap of the buck-boost transformer as connected. However the center tap will not be a grounded conductor, and will in fact be at 60V relative to ground. For some devices this makes no difference at all, because there are no exposed terminals and all supply conductors are switched. But anything that depends upon the neutral being a grounded conductor could be a hazard. Using this 'derived neutral' is probably a code violation.
If you attempt to connect the center tap of the transformer to the grounded neutral of the wye source, you will see lots of 'circulating current' in the transformer set. Without the center tap connected to the grounded neutral, you only have a pair of connected terminals, and thus a single phase connected to the buck-boost transformer. But with the center tap connected, you have two phases with a real phase angle difference. These two phases drive two separate 120V coils, but the two coils are on the same magnetic core. The net result is that the magnetic flux induced in the core will not be in phase with the voltage applied to the coils, and current will flow to compensate. Another way to look at this is that current will flow to reconcile the fact that the buck-boost transformer is trying to hold the 'center tap' at 60V relative to the grounded neutral that it is connected to.
In the original drawing, assuming a 208/120V wye source, the voltage between X1 and X4 is about 236V, and the voltage between X1 (or X4) and ground is about 132V. If instead you connect both of the X coils on one 'side' of the transformer, and just use one of the supply legs directly as a supply to your 240V load, then that leg will also be at 120V to ground. The _other_ leg, however will be at 145V to ground. If you know which leg serves both 120V and 240V loads, then you could use a buck-boost transformer to supply a 120/240V load from a 208/120V wye system. Of course if you get this wrong than the 120V components will be significantly overloaded.
-Jon