I have characters who have to use false names.
Should the narrator use their real name or false name?
For example, if person A was being called Adam when their real name was Archie, person B would call them Adam, but should the narrator say "Adam walked over" or "Archie walked over"?
I have characters who have to use false names.
Use the real name.
I do this all the time, in my current writing the main character (a female) is often spying to gain information, and pretends to be a fictitious person to do it.
My narrator always calls her by her real name. She introduces herself by her assumed name and responds only to it. She never uses her real name anywhere but in her real life.
"Karen?" Jon asked.
Ariel looked up. "Yes sir?"
I have not written a situation in which overlap between missions occurs; that would be unusual in my story universe, but if I did, she would make an excuse to any person that knew her by a previous name. For example, that in a previous job she had stopped using "Mary" and taken to using her middle name "Karen" to avoid a conflict with another "Mary". Something like that.
The narrator should normally use the real name and be consistent with the usage.
Have a look at the question Would it be cheating to change the main character's “name” partway through the story? for an idea of how changing the name midway through the story can be done and discussions about whether this is a good idea or not. But note that the author in that case only wants to switch the name once.
If the narrator should know about the real name he should normally use that one. You can then explain what each fake name means that Archie created and why certain people are calling him Adam. Especially if you are planning to use multiple fake names, such as Archie being called Adam by people in his school and Bert by people in his sports club and Dean when talking with strangers on the internet.
Comparing it to your username might help you to visualize this. My friends and family wouldn't call me Secespitus in real life. Even if they know this name that I have given myself as a sort of fake identity, it wouldn't occur to them to use this name if there was not a compelling reason to do so.
Real name > Fake name
That is true if and only if the narrator is supposed to know the real name of the character. For instance, in a first person narrative, the narrator may not be privy to such information, hence they will be stuck with the name that they think to be "real". If the narrator discovers the true name of the character during the story, they may decide to switch for the true name, or keep the fake one, given that they have become accustomed to it.
I had the same issue in one of my stories. In it, two female ninjas were infiltrating the dregs of society in a port city, and needed to go by pseudonyms (it wasn't a serious training according to the point of view character, and she has a major sass problem).
So what did I do? I put the question to the character. Because I was writing 1st person (my usual thing, though this works well when writing in 3rd person, if close to a sole character. Be warned that if you switch the lens, you need to switch the name, and that gets confusing). So Aya, the point of view character, brought her usual sass to bear, and she referred to her companion as 'Kara' (seriously, with the quotations every time) for the duration of their infiltration.
My advice to you, is look at what matches the narrative. If written in omniscient 3rd person, you can't really get away with much, but in 3rd limited and 1st? You can use the voice of your point of view character to tell you what makes more sense in the situation. Spy constantly changing name? They might need to think of themselves by this name to keep from coming over as apathetic to the name or disingenuous. Sassy character that hates the situation, but understands it well enough to cooperate? Count on taking little victories whenever they can.
If it is your MC that's using a pseudonym, how do they think of themselves? Have they become the mask, adopted a new personality, put their real identity in a closed box? Or do they sometimes forget to respond when called by their pseudonym? You can convey information by the way your narrator calls your characters. For example, in his "Belgariad" series, David Eddings always refers to his MC as 'Garion'. The character's proper name is 'Belgarion', the 'Bel' part signifying he's a sorcerer and a person of high status in the world. By calling him 'Garion', Eddings tells us that the character is not wholly comfortable with the whole position and power, not quite used to it yet. In the same book series, another character is referred by the narrator as 'Silk'. 'Silk' is a pseudonym of a prince, who lives the life of a spy. Spying is his life, what he does and enjoys doing. Being a prince is, to this character, something random that happened. He prefers being 'Silk' to being 'Prince Kheldar'.
If it's not the MC that's living under a pseudonym, how was the pseufonymed character first introduced to them - under the fake name, or the real one? Does the MC continue thinking of them under one name, having to remind themselves to use the other?