  We introduce real second-order freeness in second-order noncommutative
probability spaces. We demonstrate that under this definition, three real
models of random matrices, namely real Ginibre matrices, Gaussian orthogonal
matrices, and real Wishart matrices, are asymptotically second-order free.
These ensembles do not satisfy the complex definition of second-order freeness
satisfied by their complex analogues. We use a combinatorial approach to the
matrix calculations similar to the genus expansion for complex random matrices,
but in which nonorientable surfaces appear, demonstrating the commonality
between the real models and the distinction from their complex analogues,
motivating this distinct definition. In the real case we find, in addition to
the terms appearing in the complex case corresponding to annular spoke
diagrams, an extra set of terms corresponding to annular spoke diagrams in
which the two circles of the annulus are oppositely oriented, and in which the
matrix transpose appears.
