We investigate real eigenvalues of real elliptic Ginibre matrices of size n , indexed by the parameter of asymmetry

τ \in [0, 1]

. In both the strongly and weakly non-Hermitian regimes, where

τ \in [0, 1)

is fixed or

1 - τ = O (1 / n)

, respectively, we derive the asymptotic expansion of the probability

p_{n, n - 2 l}

that all but a finite number 2 l of eigenvalues are real. In particular, we show that the expansion is of the form

\log p_{n, n - 2 l} = {\begin{array}{l} a_{1} n^{2} + a_{2} n + a_{3} \log n + O (1) & at strongnon-Hermiticity, \\ b_{1} n + b_{2} \log n + b_{3} + o (1) & at weaknon-Hermiticity, \end{array}

and we determine all coefficients explicitly. Furthermore, in the special case where l = 1, we derive the full-order expansions. For the proofs, we employ distinct methods for the strongly and weakly non-Hermitian regimes. In the former case, we utilise potential-theoretic techniques to analyse the free energy of elliptic Ginibre matrices conditioned to have

n - 2 l

real eigenvalues, together with the strong Szegö limit theorems. In the latter case, we utilise the skew-orthogonal polynomial formalism and the asymptotic behaviour of the Hermite polynomials.

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