  A truncation of a Haar distributed orthogonal random matrix gives rise to a
matrix whose eigenvalues are either real or complex conjugate pairs, and are
supported within the closed unit disk. This is also true for a product $P_m$ of
$m$ independent truncated orthogonal random matrices. One of most basic
questions for such asymmetric matrices is to ask for the number of real
eigenvalues. In this paper, we will exploit the fact that the eigenvalues of
$P_m$ form a Pfaffian point process to obtain an explicit determinant
expression for the probability of finding any given number of real eigenvalues.
We will see that if the truncation removes an even number of rows and columns
from the original Haar distributed orthogonal matrix, then these probabilities
will be rational numbers. Finally, based on exact finite formulae, we will
provide conjectural expressions for the asymptotic form of the spectral density
and the average number of real eigenvalues as the matrix dimension tends to
infinity.
