I didn't read any further than the abstract but that is one of the key issues that we answer philosophically rather than through science. What does "real" mean really mean?
She defines two definitions of what she means by "real":
Quote from Barrett
Natural sciences like physics deal with scientific categories that are assumed to be observer independent (they are real in the natural sense and can be discovered by humans)
Social sciences like sociology or economics deal with categories that are observer dependent (and are real because they are invented and shared by humans).
I think those are good basic definitions of how we use "real" in conversation. And Epicurus advised using definitions that could be agreed on by the average person. Photosynthesis - a natural process discoverable by science - is "real" in the first sense. Money - a culturally agreed-upon system of commerce - is "real" in the second sense.
I think your premise that philosophy is the arbiter of what's "real" is a tad restrictive unless you read her definitions as broadly "philosophical."