We study the distribution of the eigenvalue condition numbers κ i = (l * i l i )(r * i r i ) associated with real eigenvalues λ i of partially asymmetric N × N random matrices from the Gaussian elliptic ensemble. The large values of κ i signal about the non-orthogonality of (bi-orthogonal) set of left l i and right r i eigenvectors and enhanced sensitivity of the associated eigenvalues against perturbations of the matrix entries. We derive the general finite N expression for the joint probability density (JPD) P N (z, t) of t i = κ 2 i − 1 for λ conditioned to have a value z and investigate its several scaling regimes in the limit N → ∞. When the degree of asymmetry is fixed as N → ∞, the number of real eigenvalues is O( √ N ), and in the bulk of the real spectrum t i = O(N ), while on approaching the spectral edges the non-orthogonality is weaker:In both cases the corresponding JPDs, after appropriate rescaling, coincide with those found in the earlier studied case of fully asymmetric (Ginibre) matrices, see [20]. A different regime of weak asymmetry arises when a finite fraction of N eigenvalues remain real as N → ∞. In such a regime eigenvectors are weakly non-orthogonal, t = O(1), and we derive the associated JPD, finding that the characteristic tail P(z, t) ∼ t −2 survives for arbitrary weak asymmetry. As such, it is the most robust feature of the condition number density for real eigenvalues of asymmetric matrices.