It says "real" not living. So dead person would work. "Kill" is used as a verb here so can mean "to end" and you can end a corpse. Corpses are not uniformly treated as a person or an object (for instance ownership of a corpse and as property freedom of use.) I find no reason the person who's name is written must be real.
I was wondering that myself. So many questions. Do nicknames count? Like- the only way they could know would be of it was just a random person with that name, or if it was supernatural. But if it's supernatural why does it say "real person?" The existence of this supernatural force means that others are more possible too. Could you write names of supposedly mythical figures and wait to see if the door opened to prove which ones were real or not, and if it isn't supernatural, how would they know if that was a real person or not? How do you prove or disprove a person exists short of meeting them yourself? So does it have to be a person they know? How do you know who they know? If they know all the people you know and aren't a supernatural force, does that mean you know them? If you wrote their name do they kill themselves? If so does that mean if they kill themselves no one will be able to open the door? I see some issues here.
I'd call them Skippy01907651, that is now their name, as anyone can name anything and it doesn't specify legal name despite specifying real person. Thus legal name is not important just name. I'd write that on the paper and let them kill themself. The world is safe, I killed a murderer and go free, all is good in my world.
I don't want you dead, just your knees broken