The world mechanic in question is memories being part of body parts, the body parts being just a very simple to understand metaphor and visualization of the memories.
We lose memories all the time. Rather than asking which Phos is real, we could and should instead ask:>How many memories can we lose and still be ourselves?
For me, the answer is:
Internally, it doesn't matter. You always think you're yourself, even though you probably don't remember basically anything from your childhood, or even 1% of the time you spent last year. If someone or something confronts you with things you forgot, your reaction is "oh yeah, that happened". If you really, truly forgot something, then even if it's important that's the best you'd be able to do. Otherwise you didn't forget and you misjudged instead, which is a different question.
Externally, you remain who you are as long as everyone treats you that way.
We don't have to wait for a Phos' arms-and-legs episode to find out the answer in-story. We already saw it in force with Diamond and the King. In more general terms we've gotten that with a number of characters we've seen, including Phos for both limb changes.
Thinking in terms of what's "real" and what's "not real" is asking the wrong question to begin with. They're both "real".
What's important though depends on what you want out of life. Diamond doesn't act like she does in public the same way she does in private for a reason. Are you going to tell her she's wrong? That one of her is "not real"?
Would such a thing make her happy?
If you're looking for happiness, I don't think you need to ask such a question.
If you're looking for something else, then it depends what that is.