  We consider the dynamics of the non-Hermitian Su-Schrieffer-Heeger model
arising as the no-click limit of a continuously monitored free fermion chain
where particles and holes are measured on two sublattices. The model has
$\mathcal{PT}$-symmetry, which we show to spontaneously break as a function of
the strength of measurement backaction, resulting in a spectral transition
where quasiparticles acquire a finite lifetime in patches of the Brillouin
zone. We compute the entanglement entropy's dynamics in the thermodynamic limit
and demonstrate an entanglement transition between volume-law and area-law
scaling, which we characterize analytically. Interestingly we show that the
entanglement transition and the $\mathcal{PT}$-symmetry breaking do not
coincide, the former occurring when the entire decay spectrum of the
quasiparticle becomes gapped.
