  Recent developments in quantum gas microscopy open up the possibility of
real-time observation of quantum many-body systems. To understand the dynamics
of atoms under such circumstances, we formulate the dynamics under a real-time
spatially resolved measurement and show that, in an appropriate limit of weak
spatial resolution and strong atom-light coupling, the measurement
indistinguishability of particles results in complete suppression of relative
positional decoherence. As a consequence, quantum correlation in the
multi-particle dynamics persists under a minimally destructive observation. We
numerically demonstrate this for ultracold atoms in an optical lattice. Our
theoretical framework can be applied to feedback control of quantum many-body
systems which may be realized in subwavelength-spacing lattice systems.
