  We consider the complex eigenvalues of a Wishart type random matrix model
$X=X_1 X_2^*$, where two rectangular complex Ginibre matrices $X_{1,2}$ of size
$N\times (N+\nu)$ are correlated through a non-Hermiticity parameter
$\tau\in[0,1]$. For general $\nu=O(N)$ and $\tau$ we obtain the global limiting
density and its support, given by a shifted ellipse. It provides a
non-Hermitian generalisation of the Marchenko-Pastur distribution, which is
recovered at maximal correlation $X_1=X_2$ when $\tau=1$. The square root of
the complex Wishart eigenvalues, corresponding to the non-zero complex
eigenvalues of the Dirac matrix $\mathcal{D}=\begin{pmatrix} 0 & X_1 \\ X_2^* &
0 \end{pmatrix},$ are supported in a domain parametrised by a quartic equation.
It displays a lemniscate type transition at a critical value $\tau_c,$ where
the interior of the spectrum splits into two connected components. At
multi-criticality we obtain the limiting local kernel given by the edge kernel
of the Ginibre ensemble in squared variables. For the global statistics, we
apply Frostman's equilibrium problem to the 2D Coulomb gas, whereas the local
statistics follows from a saddle point analysis of the kernel of orthogonal
Laguerre polynomials in the complex plane.
