We show that pure spin injection from a magnetic electrode into an inversion symmetry-broken system composed of a tunnel barrier and a metallic region generates a transverse charge current. Such a tunneling spin galvanic conversion is robust to disorder and nonlocal, i.e., injection and detection contacts do not coincide, and is strongly anisotropic whenever the internal spin-orbit field has a nontrivial angular dependence. The anisotropy shows up in linear response, contrary to what happens in bulk conversion setups lacking tunneling elements. This is particularly relevant for spin-charge conversion at oxide interfaces, where both the tunnel barrier and the receiving low-dimensional metallic system host effective spin-orbit fields with complex angular symmetries.