We demonstrate an approach to the reconstruction of scanning probe X-ray diffraction tomography data with anisotropic polycrystalline samples. The method involves reconstructing a voxel map containing an orientation distribution function in each voxel of a bulk 3D sample. By choosing a particular kind of basis functions, we can effectively utilize non-negativity in orientation space for samples with sparse texture. This enables us to achieve stable solutions at high angular resolutions where the problem would otherwise be underdetermined. This method differs from established approaches by not relying on a peak-finding step. It is therefore applicable to sample systems consisting of small and highly mosaic crystalline domains that are not handled well by these methods. We demonstrate the new approach using data from a shot-peened martensite sample where we are able to map the twinning microstructure in the interior of a bulk sample without resolving the individual lattice domains. We also demonstrate the approach on a piece of gastropod shell with a mosaic microstructure. The results suggest that, by utilizing the sparsity of the texture, the experiment can be carried out using only a single rotation axis, unlike previous demonstrations of texture and tensor tomography.