  We discuss how quantum jumps affect localized regimes in driven-dissipative
disordered many-body systems featuring a localization transition. We introduce
a deformation of the Lindblad master equation that interpolates between the
standard Lindblad and the no-jump non-Hermitian dynamics of open quantum
systems. As a platform, we use a disordered chain of hard-core bosons with
nearest-neighbor interactions and subject to incoherent drive and dissipation
at alternate sites. We probe both the statistics of complex eigenvalues of the
deformed Liouvillian and dynamical observables of physical relevance. We show
that reducing the number of quantum jumps, achievable through realistic
postselection protocols, can promote the emergence of the localized phase. Our
findings are based on exact diagonalization and time-dependent matrix-product
states techniques.
