  In this work, we provide a comprehensive set of differential cross-section
distributions for photon + di-jet production in proton-proton collisions with
next-to-next-to-leading order precision in massless QCD. The event selection
corresponds to recent measurements by the ATLAS collaboration. We observe an
improved description of data in comparison to lower-order calculations in the
case of observables that are expected to be well described by perturbation
theory. The results also show better agreement with data than
parton-shower-matched and multi-jet-merged predictions generated for the ATLAS
analysis using the \textsc{Sherpa} Monte Carlo. A particular highlight of our
study is the use of exact five-point two-loop virtual amplitudes. This is the
first calculation of a complete two-to-three hadron-collider process at
next-to-next-to-leading order in QCD that does not rely on the leading-colour
approximation at two loops. We demonstrate, nevertheless, that the
sub-leading-colour effects present in the infrared- and ultraviolet-finite
double-virtual contributions are negligible in view of the remaining scale
uncertainties.
