Angle-resolved photoemission spectroscopy has been developed to a very high accuracy. However, effects that depend sensitively on the state of the emitted photoelectron were so far hard to compute for real molecules. We here show that the real-time propagation approach to time-dependent density functional theory allows us to obtain final-state effects consistently from first principles and with an accuracy that allows for the interpretation of experimental data. In a combined theoretical and experimental study we demonstrate that the approach captures three hallmark effects that are beyond the final-state plane-wave approximation: emission perpendicular to the light polarization, circular dichroism in the photoelectron angular distribution, and a pronounced energy dependence of the photoemission intensity.