Probability of all eigenvalues real for products of standard Gaussian matrices

With $\{X_i\}$ independent $N \times N$ standard Gaussian random matrices, the probability $p_{N,N}^{P_m}$ that all eigenvalues are real for the matrix product $P_m = X_m X_{m-1} \cdots X_1$ is expressed in terms of an $N/2 \times N/2$ ($N$ even) and $(N+1)/2 \times (N+1)/2$ ($N$ odd) determinant. The entries of the determinant are certain Meijer $G$-functions. In the case $m=2$ high precision computation indicates that the entries are rational multiples of $\pi^2$, with the denominator a power of 2, and that to leading order in $N$ $p_{N,N}^{P_m}$ decays as $(\pi/4)^{N^2/2}$. We are able to show that for general $m$ and large $N$, $p_{N,N}^{P_m} \sim b_m^{N^2}$ with an explicit $b_m$. An analytic demonstration that $p_{N,N}^{P_m} \to 1$ as $m \to \infty$ is given.