  We suggest a method of studying the joint probability density (JPD) of an
eigenvalue and the associated 'non-orthogonality overlap factor' (also known as
the 'eigenvalue condition number') of the left and right eigenvectors for
non-selfadjoint Gaussian random matrices of size $N\times N$. First we derive
the general finite $N$ expression for the JPD of a real eigenvalue $\lambda$
and the associated non-orthogonality factor in the real Ginibre ensemble, and
then analyze its 'bulk' and 'edge' scaling limits. The ensuing distribution is
maximally heavy-tailed, so that all integer moments beyond normalization are
divergent. A similar calculation for a complex eigenvalue $z$ and the
associated non-orthogonality factor in the complex Ginibre ensemble is
presented as well and yields a distribution with the finite first moment. Its
'bulk' scaling limit yields a distribution whose first moment reproduces the
well-known result of Chalker and Mehlig \cite{ChalkerMehlig1998}, and we
provide the 'edge' scaling distribution for this case as well. Our method
involves evaluating the ensemble average of products and ratios of integer and
half-integer powers of characteristic polynomials for Ginibre matrices, which
we perform in the framework of a supersymmetry approach. Our paper complements
recent studies by Bourgade and Dubach \cite{BourgadeDubach}.
