An evil surgeon has performed a dreadful experiment. The subject, whom we'll call "Holo", has had his entire nervous system transected right down the middle. Through advances in accelerated cloning, the surgeon has prepared two biological bodies, each genetially identical to Holo's original body. Into one body he transplants the left half of Holo's nervous system, and into the other goes the right half. Though evil, the surgeon is extremely skilled, and both halves of the nervous system survive and are active.
To talk about the case after the surgery, we'll use "Holo-Left" to refer to the left half of the nervous system, and "Holo-Right" to refer to the right half, each in separate bodies. Note that Holo-Left is paralyzed on the left half of his body, and similarly for Holo-Right. Both are capable of speaking, however, and except for their paralysis, and blindness in half the visual field, they do not report any serious apparent changes. That is, both Holo-Left and Holo-Right can reflect on their condition, before and after the experiment, with a clear memory, and report feeling as if they are the same person. But are they really? What has happened here? [This case assumes some pretty unlikely science, but for the sake of philosophical progress, we'll take it as given.]