  We show that the dynamics induced by post-selected measurements can serve as
a controlled route to access physical processes beyond the boundaries of
Luttinger liquid physics. We consider a one-dimensional fermionic wire whose
dynamics results from a sequence of weak measurements of the fermionic density
at a given site, interspersed with unitary hopping dynamics. This realizes a
non-Hermitian variant of the celebrated instance of a local scatterer in a
fermionic system and its ensuing orthogonality catastrophe. We observe a
distinct crossover in the system's time evolution as a function of the fermion
density. In the high-density regime, reminiscent of the Hermitian case,
Tomonaga-Luttinger liquid physics properly describes the dynamics while, as we
delve into the low-density regime, the validity of bosonization breaks down,
giving rise to irreversible behavior. Notably, this crossover from reversible
to irreversible dynamics is non-perturbative in the measurement rate and can
manifest itself even with relatively shallow measurement rates, provided that
the system's density remains below the critical threshold. Our results render a
conceptually transparent model for exploring non-perturbative effects beyond
bosonization, which could be used as a step-stone to explore novel routes for
the control of non-linear dynamics in low-dimensional quantum systems.
