  We report on a detailed investigation of the next-to-leading order (NLO) QCD
corrections to $W\gamma$+jet production at the Tevatron and the LHC using a
fully-flexible parton-level Monte Carlo program. We include the full leptonic
decay of the $W$, taking into account all off-shell and finite width effects,
as well as non-standard $WW\gamma$ couplings. We find particularly sizable
corrections for the currently allowed parameter range of anomalous couplings
imposed by LEP data. In total the NLO differential distributions reveal a
substantial phase space dependence of the corrections, leaving considerable
sensitivity to anomalous couplings beyond scale uncertainty at large momentum
transfers in the anomalous vertex.
