  We study the time evolution of long quantum spin chains subjected to
continuous monitoring via matrix product states (MPS) at fixed bond dimension,
with the Time-Dependent Variational Principle (TDVP) algorithm. The latter
gives an effective classical non-linear evolution with a conserved charge,
which approximates the real quantum evolution up to an error. We show that the
error rate displays a phase transition in the monitoring strength, which can be
well detected by scaling analysis with relatively low values of bond
dimensions. The method allows for an efficient numerical determination of the
critical measurement-induced phase transition parameters in many-body quantum
systems. Moreover, in the presence of U(1) global spin charge, we show the
existence of a charge-sharpening transition well separated from the
entanglement transition which we detect by studying the charge fluctuations of
a local sub-part of the system at very large times. Our work substantiates the
TDVP time evolution as a method to identify measured-induced phase transitions
in systems of arbitrary dimensions and sizes.
