We perform multidimensional radiative transfer simulations to compute spectra for a hydrodynamical simulation of a line-driven accretion disc wind from an active galactic nucleus. The synthetic spectra confirm expectations from parametrized models that a disc wind can imprint a wide variety of spectroscopic signatures including narrow absorption lines, broad emission lines and a Compton hump. The formation of these features is complex with contributions originating from many of the different structures present in the hydrodynamical simulation. In particular, spectral features are shaped both by gas in a successfully launched outflow and in complex flows where material is lifted out of the disc plane but ultimately falls back. We also confirm that the strong Fe Kα line can develop a weak, red-skewed line wing as a result of Compton scattering in the outflow. In addition, we demonstrate that X-ray radiation scattered and reprocessed in the flow has a pivotal part in both the spectrum formation and determining the ionization conditions in the wind. We find that scattered radiation is rather effective in ionizing gas which is shielded from direct irradiation from the central source. This effect likely makes the successful launching of a massive disc wind somewhat more challenging and should be considered in future wind simulations.