According to materialism, the conscious mind is the product of the brain. Thus, if the brain is split in half such that the two hemispheres cannot communicate with each other, then there are three possibilities:
- Both hemispheres gain their own conscious mind.
- Only one hemispheres has a conscious mind.
- A single conscious mind operates across both hemispheres.
Within a materialistic paradigm, #3 is impossible. There cannot be unified control without communication between the two hemispheres.
Option #1 is false according to this article,
A new research study contradicts the established view that so-called split-brain patients have a split consciousness. Instead, the researchers behind the study, led by UvA psychologist Yair Pinto, have found strong evidence showing that despite being characterised by little to no communication between the right and left brain hemispheres, split brain does not cause two independent conscious perceivers in one brain. Their results are published in the latest edition of the journal Brain.
and this answer.
So I conclude from [sic] this quick literature review that Joseph ascribes the presence of two minds based on something that can be explained at a more basic level, and hence by a more simple model, namely disinhibited motor control. Given the principle of parsimony, I conclude his observations were right, but that his conclusions were not.
Option #2 is false according to this answer. The answer state consciousness does not transfer to one half, but continues to operate in both halves.
If by continuity you mean, "a feeling that I am who I was before the operation, (perhaps with some changes)", then it seems that each hemisphere would separately maintain continuity, in the same way patients after massive strokes and other sudden brain injuries don't usually feel "they are a different person".
Thus, only option #3 is viable according to the available data.
If option #3 is indeed the only possible explanation, yet is impossible according to materialism, then the materialistic explanation for the conscious mind is false.
A fourth possibility is there is an alternative communication pathway that is not severed, as hinted at by this answer.
Keep in mind that there are many other pathways between the hemispheres besides the Corpus Callosum - such as thalamocortical circuits, those via the midbrain and brainstem and of course external sensorimotor feedback.
Is option #3 the only viable option, or do the halves somehow communicate in a different way?