  Random matrices from the elliptic Ginibre orthogonal ensemble (GinOE) are a
certain linear combination of a real symmetric, and real anti-symmetric, real
Gaussian random matrices and controlled by a parameter $\tau$. Our interest is
in the fluctuations of the number of real eigenvalues, for fixed $\tau$ when
the expected number is proportional to the square root of the matrix size $N$,
and for $\tau$ scaled to the weakly non-symmetric limit, when the number of
eigenvalues is proportional to $N$. By establishing that the generating
function for the probabilities specifying the distribution of the number of
real eigenvalues has only negative real zeros, and using too the fact that
variances in both circumstancesof interest tends to infinity as $N \to \infty$,
the known central limit theorem for the fluctuations is strengthened to a local
central limit theorem, and the rate of convergence is discussed.
