Your superhero persona and your mundane persona have built a wall in between them, in your head. They behave like two distinct people. They can have different moral systems, love different people, and see the world from completely different points of view. Your personae might have different skills, feats, or even different classes (ability scores and powers remain the same). Neither remembers the experiences of the other, but both will invent reasons to seek a private space to change personae (and clothes). Neither persona needs to be “fake,” necessarily. They could both be very real, honest parts of your personality. They’re just split from each other. If one of them is fake, it doesn’t have to be the mundane one. The superhero persona might have sought refuge in an invented “normal” life.
Normally, if you’re in your mundane persona and you take damage from an attack or are forced to roll a saving throw, you will switch to your superhero persona to protect yourself. When you finish a fight or mission as your superhero persona, you will change back to your mundane persona within 1d4 minutes. Also, if you go to sleep as your superhero persona, you will invariably wake up as your mundane persona, even if you’ve been rendered unconscious in a fight and wake up in that same fight.
You can voluntarily switch between personae or resist switching personae. Both require a Will check (DC 20). The GM might raise this DC under extreme circumstances, for example, if you’re being shot at but want to stay as your mundane persona, or if a loved one is pleading with your superhero persona to change back into your mundane self.
If you combine this Comp with Form Shift, then each personality will correspond to one form. Usually, this means that your empowered form will correspond with your heroic persona. If you combine the two, your ability scores might shift between personae because you lose your powers in mundane persona form.
It is entirely possible to buy off this Comp, perhaps even as the culmination of “years” of hard work on the part of the character. When that happens, the two personalities integrate with each other, becoming a combination of the two, a middle-ground that remembers everything that both experienced.