  Let D^n be the closed unit polydisk in C^n. Consider the ring C_r of
complex-valued continuous functions on D^n that are real symmetric, that is,
f(z)=(f(z^*))^* for all z in D^n. It is shown that C_r is projective free, that
is, finitely generated projective modules over C_r are free. We also show that
several subalgebras of the real symmetric polydisc algebra are projective free.
