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The probability that all eigenvalues of a product of $m$ independent $N\times N$ sub-blocks of a Haar distributed random real orthogonal matrix ofsize $(L_i+N) \times (L_i+N)$, $(i=1,\dots,m)$ are real is calculated as amulti-dimensional integral, and as a determinant. Both involve MeijerG-functions. Evaluation formulae of the latter, based on a recursive scheme,allow it to be proved that for any $m$ and with each $L_i$ even the probabilityis a rational number. The formulae furthermore provide for explicit computationin small order cases.

The Probability That All Eigenvalues are Real for Products of Truncated Real Orthogonal Random Matrices