The paper investigates the time evolution of long quantum spin chains under continuous monitoring using matrix product states (MPS) and the Time-Dependent Variational Principle (TDVP) algorithm. The TDVP algorithm provides an effective classical non-linear evolution with a conserved charge, which approximates the real quantum evolution with an error rate that displays a phase transition in the monitoring strength. This phase transition can be detected through scaling analysis with relatively low bond dimensions. The method is efficient for numerically determining critical measurement-induced phase transition parameters in many-body quantum systems. Additionally, the presence of a $U(1)$ global spin charge allows for the detection of a charge-sharpening transition, which occurs at a smaller measurement rate compared to the entanglement transition. The authors demonstrate the effectiveness of their approach by studying the entanglement and charge-sharpening transitions in both the XXX and J-XXX models, showing that the critical parameters are accurately extracted even with small bond dimensions. The findings provide a robust method for identifying measured-induced phase transitions in systems of arbitrary dimensions and sizes.The paper investigates the time evolution of long quantum spin chains under continuous monitoring using matrix product states (MPS) and the Time-Dependent Variational Principle (TDVP) algorithm. The TDVP algorithm provides an effective classical non-linear evolution with a conserved charge, which approximates the real quantum evolution with an error rate that displays a phase transition in the monitoring strength. This phase transition can be detected through scaling analysis with relatively low bond dimensions. The method is efficient for numerically determining critical measurement-induced phase transition parameters in many-body quantum systems. Additionally, the presence of a $U(1)$ global spin charge allows for the detection of a charge-sharpening transition, which occurs at a smaller measurement rate compared to the entanglement transition. The authors demonstrate the effectiveness of their approach by studying the entanglement and charge-sharpening transitions in both the XXX and J-XXX models, showing that the critical parameters are accurately extracted even with small bond dimensions. The findings provide a robust method for identifying measured-induced phase transitions in systems of arbitrary dimensions and sizes.