  We demonstrate the asymptotic real second order freeness of Haar distributed
orthogonal matrices and an independent ensemble of random matrices. Our main
result states that if we have two independent ensembles of random matrices with
a real second order limit distribution and one of them is invariant under
conjugation by an orthogonal matrix, then the two ensembles are asymptotically
real second order free. This captures the known examples of asymptotic real
second order freeness introduced by Redelmeier [R1, R2].
