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 utilize 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 cou-
pling regime. This work opens opportunities to understand continuous monitoring of non-Markovian open
quantum systems, both on a practical and fundamental level