  We show how the replica method can be used to compute the asymptotic
eigenvalue spectrum of a real Wishart product matrix. For unstructured factors,
this provides a compact, elementary derivation of a polynomial condition on the
Stieltjes transform first proved by M\"uller [IEEE Trans. Inf. Theory. 48,
2086-2091 (2002)]. We then show how this computation can be extended to
ensembles where the factors are drawn from matrix Gaussian distributions with
general correlation structure. For both unstructured and structured ensembles,
we derive polynomial conditions on the average values of the minimum and
maximum eigenvalues, which in the unstructured case match the results obtained
by Akemann, Ipsen, and Kieburg [Phys. Rev. E 88, 052118 (2013)] for the complex
Wishart product ensemble.
