  We consider the elliptic Ginibre matrices in the orthogonal symmetry class
that interpolates between the real Ginibre ensemble and the Gaussian orthogonal
ensemble. We obtain the finite size corrections of the real eigenvalue
densities in both the global and edge scaling regimes, as well as in both the
strong and weak non-Hermiticity regimes. Our results extend and provide the
rate of convergence to the previous recent findings in the aforementioned
limits. In particular, in the Hermitian limit, our results recover the finite
size corrections of the Gaussian orthogonal ensemble established by Forrester,
Frankel and Garoni.
