  We consider non-Gaussian extensions of the elliptic Ginibre ensemble of
complex non-Hermitian random matrices by fixing the trace
$\operatorname{Tr}(XX^*)$ of the matrix $X$ with a hard or soft constraint.
These ensembles have correlated matrix entries and non-determinantal joint
densities of the complex eigenvalues. We study global and local bulk statistics
in these ensembles, in particular in the limit of weak non-Hermiticity
introduced by Fyodorov, Khoruzhenko and Sommers. Here, the support of the
limiting measure collapses to the real line. This limit was motivated by
physics applications and interpolates between the celebrated sine and Ginibre
kernel. Our results constitute a first proof of universality of the
interpolating kernel. Furthermore, in the limit of strong non-Hermiticity,
where the support of the limiting measure remains an ellipse, we obtain local
Ginibre statistics in the bulk of the spectrum.
