  We derive an accurate lower tail estimate on the lowest singular value
$\sigma_1(X-z)$ of a real Gaussian (Ginibre) random matrix $X$ shifted by a
complex parameter $z$. Such shift effectively changes the upper tail behaviour
of the condition number $\kappa(X-z)$ from the slower
$\mathbf{P}(\kappa(X-z)\ge t)\lesssim 1/t$ decay typical for real Ginibre
matrices to the faster $1/t^2$ decay seen for complex Ginibre matrices as long
as $z$ is away from the real axis. This sharpens and resolves a recent
conjecture in [arXiv:2005.08930] on the regularizing effect of the real Ginibre
ensemble with a genuinely complex shift. As a consequence we obtain an improved
upper bound on the eigenvalue condition numbers (known also as the eigenvector
overlaps) for real Ginibre matrices. The main technical tool is a rigorous
supersymmetric analysis from our earlier work [arXiv:1908.01653].
