  Using large $N$ arguments, we propose a scheme for calculating the two-point
eigenvector correlation function for non-normal random matrices in the large
$N$ limit. The setting generalizes the quaternionic extension of free
probability to two-point functions. In the particular case of biunitarily
invariant random matrices, we obtain a simple, general expression for the
two-point eigenvector correlation function, which can be viewed as a further
generalization of the single ring theorem. This construction has some striking
similarities to the freeness of the second kind known for the Hermitian
ensembles in large $N$. On the basis of several solved examples, we conjecture
two kinds of microscopic universality of the eigenvectors - one in the bulk,
and one at the rim. The form of the conjectured bulk universality agrees with
the scaling limit found by Chalker and Mehlig [JT Chalker, B Mehlig, PRL,
\textbf{81}, 3367 (1998)] in the case of the complex Ginibre ensemble.
