There have been plenty of cases where hemispheres split. It doesn't present in separate consciousnesses. That's due to the fact that there are plenty of structures in the brain that aren't "Split" such as the brain stem, and prefrontal cortex.
I've also met a therapist (physical) who lost nearly 40% of their brain by volume. Due to a stroke. I've had a stroke as well, which is why I was in physical rehab as well. Consciousness can remain intact (for the most part) through the most horrific injuries to the brain.
The prefrontal cortex is the [i]executive[/i] center of the brain and acts generally as a unified whole. Its the part that lets you plan or perceive outcomes of various plans. THEN afterwards it access either hemisphere for various subtasks related to planning.
More of an anecdotal opinion on the matter ,but consciousness is still a rather mysterious thing. They, scientists, suggest that it might be an emergent phenomena of the whole totality of the brain.