Diffusion-weighted imaging (DWI) facilitates probing neural tissue structure non-invasively by measuring its hindrance to water diffusion. Analysis of DWI is typically based on generative signal models for given tissue geometry and microstructural properties. In this work, we generalize multi-tissue spherical deconvolution to a blind source separation problem under convexity and nonnegativity constraints. This spherical factorization approach decomposes multi-shell DWI data, represented in the basis of spherical harmonics, into tissue-specific orientation distribution functions and corresponding response functions, without assuming the latter as known thus fully unsupervised. In healthy human brain data, the resulting components are associated with white matter fibres, grey matter, and cerebrospinal fluid. The factorization results are on par with state-of-the-art supervised methods, as demonstrated also in Monte-Carlo simulations evaluating accuracy and precision of the estimated response functions and orientation distribution functions of each component. In animal data and in the presence of oedema, the proposed factorization is able to recover unseen tissue structure, solely relying on DWI. As such, our method broadens the applicability of spherical deconvolution techniques to exploratory analysis of tissue structure in data where priors are uncertain or hard to define.