The probability that all eigenvalues of a product of m independent N× N subblocks of a Haar distributed random real orthogonal matrix of size (Li+ N) × (Li+ N) , (i= 1 , ⋯ , m) are real is calculated as a multidimensional integral, and as a determinant. Both involve Meijer G-functions. Evaluation formulae of the latter, based on a recursive scheme, allow it to be proved that for any m and with each Li even the probability is a rational number. The formulae furthermore provide for explicit computation in small order cases. © 2017, Springer Science+Business Media New York.