  We introduce a family of matrices with non-commutative entries that
generalize the classical real Wishart matrices.
  With the help of the Brauer product, we derive a non-asymptotic expression
for the moments of traces of monomials in such matrices; the expression is
quite similar to the formula derived in our previous work for independent
complex Wishart matrices. We then analyze the fluctuations about the
Marchenko-Pastur law. We show that after centering by the mean, traces of real
symmetric polynomials in q-Wishart matrices converge in distribution, and we
identify the asymptotic law as the normal law when q=1, and as the semicircle
law when q=0.
