  Quantum measurements profoundly influence system dynamics. They lead to
complex nonequilibrium phenomena like the quantum Zeno effect, and they can be
used for mitigating errors in quantum simulations.
  Such an ability is particularly valuable for lattice gauge theories (LGTs),
which require the challenging preservation of an extensive number of local
conservation laws.
  While it is known that tailored quantum measurements can soften violations of
gauge symmetry, the nature of this protection, and in particular the
possibility of a threshold behavior, is still unexplored.
  Here, we demonstrate the existence of a sharp transition, triggered by the
measurement rate, between a protected gauge-theory regime resistant to
simulation errors and an irregular regime.
  Our results are based on the paradigmatic example of a 1+1d $\mathbb{Z}_2$
LGT. We study in detail the protection through projective measurements of
ancillary qubits coupled to the local symmetry generators, and compare this
approach with analog (weak) measurement protocols.
  We show that, while the resulting ensemble averages in the continuous-time
limit share the same Liouvillian dynamics, different physical implementations
of the stochastic gauge protection protocol yield trajectory unravelings with
vastly different statistics.
  Additionally, we design an on-chip feedback mechanism that corrects bit-flip
errors and significantly enhances the discrete-time scheme. Our results shed
light on the dissipative criticality of strongly-interacting,
highly-constrained quantum systems, and they offer valuable insights into error
mitigation and correction of gauge-theory quantum simulations.
