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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? ...
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#4: Attribution notice removed
Source: https://writers.stackexchange.com/a/33078 License name: CC BY-SA 3.0 License URL: https://creativecommons.org/licenses/by-sa/3.0/
#3: Attribution notice added
Source: https://writers.stackexchange.com/a/33078 License name: CC BY-SA 3.0 License URL: https://creativecommons.org/licenses/by-sa/3.0/
#2: Initial revision
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?](https://writing.stackexchange.com/questions/33051/would-it-be-cheating-to-change-the-main-characters-name-partway-through-the-s) 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.