  Sensitivity of an eigenvalue $\lambda_i$ to the perturbation of matrix
elements is controlled by the eigenvalue condition number defined as $\kappa_i
= \sqrt{\left< L_i | L_i\right> \left< R_i|R_i \right> }$, where
$\left<L_i\right|$ and $\left|R_i\right>$ are left and right eigenvectors to
the eigenvalue $\lambda_i$. In Random Matrix Theory the squared eigenvalue
condition number is also known as the eigenvector self-overlap. In this work we
calculate the asymptotics of the joint probability density function of the real
eigenvalue and the square of the corresponding eigenvalue condition number for
the real elliptic ensemble in the double scaling regime of almost Hermiticity
and close to the edge of the spectrum. As a byproduct, we also calculate the
one-parameter deformation of the Scorer's function.
