We extend the theory of Kassam et al. [J. Opt. Soc. Am. A12, 2009 (1995)JOAOD60740-323210.1364/JOSAA.12.002009] for scattering by oblique columnar structure thin films to include the induced form birefringence and the propagation of radiation in those films. We generalize the 4×4 matrix theory of Berreman [J. Opt. Soc. Am.62, 502 (1972)JOSAAH0030-394110.1364/JOSA.62.000502] to include arbitrary sources in the layer, which are necessary to determine the Green function for the inhomogeneous wave equation. We further extend first-order vector perturbation theory for scattering by roughness in the smooth surface limit, when the layer is anisotropic. Scattering by an inhomogeneous medium is approximated by a distorted Born approximation, where effective medium theory is used to determine the effective properties of the medium, and strong fluctuation theory is used to determine the inhomogeneous sources. In this manner, we develop a model for scattering by inhomogeneous films, with anisotropic correlation functions. The results are compared with Mueller matrix bidirectional scattering distribution function measurements for a glancing-angle deposition (GLAD) film. While the results are applied to the GLAD film example, the development of the theory is general enough that it can guide simulations for scattering in other anisotropic thin films.