  In non-Hermitian random matrix theory there are three universality classes
for local spectral correlations: the Ginibre class and the nonstandard classes
$\mathrm{AI}^\dagger$ and $\mathrm{AII}^\dagger$. We show that the continuum
Dirac operator in two-color QCD coupled to a chiral $\mathrm{U}(1)$ gauge field
or an imaginary chiral chemical potential falls in class $\mathrm{AI}^\dagger$
($\mathrm{AII}^\dagger$) for fermions in pseudoreal (real) representations of
$\mathrm{SU}(2)$. We introduce the corresponding chiral random matrix theories
and verify our predictions in lattice simulations with staggered fermions, for
which the correspondence between representation and universality class is
reversed. Specifically, we compute the complex eigenvalue spacing ratios
introduced recently. We also derive novel spectral sum rules.
