It’s really dependent on the type of community you have; in the end it’s up to you 
Most B2B communities I’ve seen have a convention of using “real” names. Since many of these communities will have people meet in real life, on linkedin etc.
Medical communities often explicitly forbid using real names for users, since that brings risks about identification.
B2C communities often allow a mix.
Here on Discourse Meta it’s also a mix, I have many colleagues that I truly only know by their handle. E.g. @dax is just “Dax” when we meet up in real life too even though I know her name is Daniela. Takes a while to get used to, but it works for us
Edit: I just misspelled Daniela’s name.. kinda proving the point. Sorry Dax! 