The efficient readout of the relevant information is pivotal for quantum simulation experiments. Often only single observables are accessed by performing standard projective measurements. In this work, we implement an atomic beam splitter by controlled outcoupling that enables a generalized measurement scheme. This gives us simultaneous access to number imbalance and relative phase in a system of two tunnel-coupled 1D Bose gases, which realize a quantum simulator of the sine-Gordon field theory. We demonstrate that the scheme is quantum limited by accessing number squeezing as a sensitive probe for added noise; further, we show that we can track Josephson oscillation dynamics with the generalized measurements. Finally, we show that the scheme allows the extraction of atoms while maintaining the system's coherent dynamics, which opens up the door to accessing multitime correlation functions. Our scheme constitutes a step towards accessing quantum properties of the sine-Gordon field theory and, in the future, studying spatially extended systems under continuous monitoring.