  We present the two-loop leading colour QCD helicity amplitudes for the
process $pp\to W(\to l\nu)\gamma+j$. We implement a complete reduction of the
amplitudes, including the leptonic decay of the $W$-boson, using finite field
arithmetic, and extract the analytic finite remainders using a recently
identified basis of special functions. Simplified analytic expressions are
obtained after considering permutations of a rational kinematic parametrisation
and multivariate partial fractioning. We demonstrate efficient numerical
evaluation of the two-loop colour and helicity summed finite remainders for
physical kinematics, and hence the suitability for phenomenological
applications.
