  In this work, we investigate the localization properties of a paradigmatic
model, coupled to a monitoring environment and possessing a many-body localized
(MBL) phase. We focus on the post-selected no-click limit with quench random
rates, i.e., random gains and losses. In this limit, the system is modeled by
adding an imaginary random potential, rendering non-Hermiticity in the system.
Numerically, we provide an evidence that the system is localized for any finite
amount of disorder. To analytically understand our results, we extend the
quantum random energy model (QREM) to the non-Hermitian scenario. The Hermitian
QREM has been used previously as a benchmark model for MBL. The QREM exhibits a
size-dependent MBL transition, where the critical value scales as $W_c\sim
\sqrt{L} \ln{L}$ with system size and presenting many-body mobility edges. We
reveal that the non-Hermitian QREM with random gain-loss offers a significantly
stronger form of localization, evident in the nature of the many-body mobility
edges and the value for the transition, which scales as $W_c\sim \ln^{1/2}{L}$
with the system size.
