The tensor tomography problem asks to what extend tensors on a manifold are determined by their integrals along geodesics. It appears as linearisation of several important nonlinear inverse problems (such as boundary or spectral rigidity) and, at least in two dimensions, it is now well understood in a variety of settings. In the talk we will consider closed Anosov surfaces and explain how the problem boils down to producing sufficiently many distributions that are invariant under the geodesic flow. A common trick to obtain more and more invariant distributions is to multiply them and, given that we are dealing with distributions, it was long assumed that this is a touchy business. Quite recently it was observed that the relevant multiplications can be performed irrespective of the underlying geometry, which greatly simplifies the solution of the tensor tomography problem. The last bit of the talk will explain this new proof of the multiplication property, which appears in joint work with Thibault Lefeuvre and Gabriel Paternain.