Easy.. Everyone is a mix of personalities acquired from people you spend time with and interact. It's way more than just 2. Each side might hold a huge chunk of partial information.

When they split, it's maybe two combination of personalities fighting each other. And each subconscious would create a different individual

But the real you can only be the complete mind where all emerges come from. I'm no expert but sounds like the soul would be a confused one and not a broken one.