  An important challenge in non-Markovian open quantum systems is to understand
what information we gain from continuous measurement of an output field. For
example, atoms in multimode cavity QED systems provide an exciting platform to
study many-body phenomena in regimes where the atoms are strongly coupled
amongst themselves and with the cavity, but the strong coupling makes it
complicated to infer the conditioned state of the atoms from the output light.
In this work we address this problem, describing the reduced atomic state via a
conditioned hierarchy of equations of motion, which provides an exact
conditioned reduced description under monitoring (and continuous feedback). We
utilise this formalism to study how different monitoring for modes of a
multimode cavity affects our information gain for an atomic state, and to
improve spin squeezing via measurement and feedback in a strong coupling
regime. This work opens opportunities to understand continuous monitoring of
non-Markovian open quantum systems, both on a practical and fundamental level.
