  The probability that there are $k$ real eigenvalues for an $n$ dimensional
real random matrix is known. Here we study this for the case of products of
independent random matrices. Relating the problem of the probability that the
product of two real 2 dimensional random matrices has real eigenvalues to an
issue of optimal quantum entanglement, this is fully analytically solved. It is
shown that in $\pi/4$ fraction of such products the eigenvalues are real. Being
greater than the corresponding known probability ($1/\sqrt{2}$) for a single
matrix, it is shown numerically that the probability that {\it all} eigenvalues
of a product of random matrices are real tends to unity as the number of
matrices in the product increases indefinitely. Some other numerical
explorations, including the expected number of real eigenvalues is also
presented, where an exponential approach of the expected number to the
dimension of the matrix seems to hold.
