  We consider complex, weakly non-Hermitian matrices $A = W_1
+i\sqrt{{\tau}_N}W_2$ , where $W_1$ and $W_2$ are Hermitian matrices and
$\tau_N = O(N^{-1})$. We first show that for pairs of Hermitian matrices $(W_1
, W_2)$ such that $W_1$ satisfies a multi-resolvent local law and $W_2$ is
bounded in norm, the bulk correlation functions of the weakly non-Hermitian
Gauss-divisible matrix $A + \sqrt{t}B$ converge pointwise to a universal limit
for $t = O(N^{-1+\epsilon})$. Using this and the reverse heat flow we deduce
bulk universality in the case when $W_1$ and $W_2$ are independent Wigner
matrices with sufficiently smooth density.
