Unity of apperception and physicalism, again

Is the unity of apperception after all consistent with event level dualism?

If it is the whole brain that thinks snow is white and grass is green, though the first be correlated with - though not identical to - one brain event and the second quite another, that appears to save the unity of apperception, to be consistent with or even in some sense explain both thoughts being thoughts of the same thinker.

If instead it is suggested that one brain event in a single brain (yours, say) is itself also a thinking that snow is white while another is a thinking that grass is green, that seems to leave open the possibility that these two thinkings could belong to different thinkers, contrary to the fact.

But experience seems to indicate that at least in the supposed cases of multiple personality different thinkings in the same brain really might belong to different thinkers.

Might it not be that whether or not multiple thinkings in the same brain belong to the same subject is contingent on any number of currently unknown physical circumstances that usually obtain but in a rare minority of people do not work out?

Given this, for the mental natures of these events to play any sort of role in the life of the organism it would seem to be necessary that events misbehave, relative to what their physical natures would alone determine.

It seems incredible that experience could be utterly gratuitous and irrelevant.

And that seems the only alternative.

Afterthought.

Of course, the physical event that happens to be also a thinking that grass is green is doubtless actually a composite of many other brain events, possibly some simultaneous and others successive.

It would be the whole composite that makes up the thinking that grass is green.