Better yet, was he real, but written about falsely?
Plato is not our our only source for the existence of Socrates. For example he appears as a character in Aristophane's The Clouds. That being said, the earlier Platonic dialogs are generally assumed to be an account of Socratic teachings.
Socrates was real. Plato, Xenophon and Aristophanes all write about him. It is difficult to tell if Plato's writings and beliefs genuinely represent the beliefs of Socrates because they could differ as much as Aristotle's does from Plato (Socrates taught Plato, who in turn taught Aristotle).
The Stanford Encyclopedia of Philosophy's article on Socrates goes into detail about Xenophon and Aristophanes writing about Socrates and touches on what seems to be be the second part of your question which is distinguishing Plato presenting Socrates as he remembers the person Socrates and Socrates the character who is a mouthpiece for Plato.