  The competition between unitary quantum dynamics and dissipative stochastic
effects, as emerging from continuous-monitoring processes, can culminate in
measurement-induced phase transitions. Here, a many-body system abruptly
passes, when exceeding a critical measurement rate, from a highly entangled
phase to a low-entanglement one. We consider a different perspective on
entanglement phase transitions and explore whether these can emerge when the
measurement process itself is modified, while keeping the measurement rate
fixed. To illustrate this idea, we consider a noninteracting fermionic system
and focus on diffusive detection processes. Through extensive numerical
simulations, we show that, upon varying a suitable \textit{unraveling
parameter} -- interpolating between measurements of different quadrature
operators -- the system displays a transition from a phase with area-law
entanglement to one where entanglement scales logarithmically with the system
size. Our findings may be relevant for tailoring quantum correlations in noisy
quantum devices and for conceiving optimal classical simulation strategies.
