  The probability that all eigenvalues of a product of $m$ independent $N
\times N$ sub-blocks of a Haar distributed random real orthogonal matrix of
size $(L_i+N) \times (L_i+N)$, $(i=1,\dots,m)$ are real is calculated as a
multi-dimensional 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 $L_i$ even the probability
is a rational number. The formulae furthermore provide for explicit computation
in small order cases.
