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

Real Second-Order Freeness and the Asymptotic Real Second-Order Freeness of Several Real Matrix Models