  We study free fermion systems under adaptive quantum dynamics consisting of
unitary gates and projective measurements followed by corrective unitary
operations. We further introduce a classical flag for each site, allowing for
an active or inactive status which determines whether or not the unitary gates
are allowed to apply. In this dynamics, the individual quantum trajectories
exhibit a measurement-induced entanglement transition from critical to area-law
scaling above a critical measurement rate, similar to previously studied models
of free fermions under continuous monitoring. Furthermore, we find that the
corrective unitary operations can steer the system into a state characterized
by charge-density-wave order. Consequently, an additional phase transition
occurs, which can be observed at both the level of the quantum trajectory and
the quantum channel. We establish that the entanglement transition and the
steering transition are fundamentally distinct. The latter transition belongs
to the parity-conserving (PC) universality class, arising from the interplay
between the inherent fermionic parity and classical labelling. We demonstrate
both the entanglement and the steering transitions via efficient numerical
simulations of free fermion systems, which confirm the PC universality class of
the latter.
