  We present a genus expansion-type expression for the expected values of
products of traces of expressions involving Haar-distributed orthogonal
matrices. As with other real genus expansions, nonorientable surfaces appear,
in addition to the orientable surfaces of the complex expansion.
  We use this expression to demonstrate that independent random matrices which
are orthogonally in general position, such as matrices whose distributions are
orthogonally invariant, are asymptotically real second-order free.
