  Random matrices formed from i.i.d. standard real Gaussian entries have the
feature that the expected number of real eigenvalues is non-zero. This property
persists for products of such matrices, independently chosen, and moreover it
is known that as the number of matrices in the product tends to infinity, the
probability that all eigenvalues are real tends to unity. We quantify the
distribution of the number of real eigenvalues for products of finite size real
Gaussian matrices by giving an explicit Pfaffian formula for the probability
that there are exactly $k$ real eigenvalues as a determinant with entries
involving particular Meijer G-functions. We also compute the explicit form of
the Pfaffian correlation kernel for the correlation between real eigenvalues,
and the correlation between complex eigenvalues. The simplest example of these
- the eigenvalue density of the real eigenvalues - gives by integration the
expected number of real eigenvalues. Our ability to perform these calculations
relies on the construction of certain skew-orthogonal polynomials in the
complex plane, the computation of which is carried out using their relationship
to particular random matrix averages.
