I am interested in finding examples (and more optimistically, a full classification) of hypergeometric functions $_2F_1(a,b;c;x)$, that evaluate to a phase (i.e. they have constant modulus) for x real and semipositive, $x\in [0,+\infty)$. A trivial example is $_2F_1(0,b;c;x)=1$. A second example is $_2F_1(-i m,b;b;-x)=(1+x)^{i m}$ for $m$ real.
My question is whether there are other examples.