I'm not sure Murph knew his real name. Murph didn't know his self. He was too conscious of his ego, and rightly so. Murph was not about to lose his ego. So Murph thought of himself as just Murph, as Paul, I suppose, had always thought he was Saul, and Peter insisted that he was Simon.
Names are strange things.
Note how we introduce ourselves, or how others refer to us, or our nicknames or names we have for friends and lovers. Well, Murph called himself Murph because he saw himself as a murph, and a very ordinary murph, while in fact he was extraordinary. Ego is he whom most people related to when they related to Murph. That's why he ended up in the desert. Murph wanted to know himself, but not at the expense of his ego.