What would happen if a Pope were to have Dissociative Identity Disorder, so that at times the Papacy was in the hands of a personality that was not competent, not the personality selected, or even not Catholic? Would the original Papal personality take it upon himself to abdicate? Could the College of Cardinals remove the Pope from office (I understand there is no precedent for this, but...)? Would the personality that wasn't originally elected Pope be considered to have any Papal standing at all?
ETA: The Pope didn't start out crazy; the DID comes from Applied Psychological/Supernatural Phlebotinum.