We describe the results of studies of a random‐anisotropy Blume‐Emery‐Griffiths spin‐1 Ising model using mean‐field theory, transfer‐matrix calculations, and position‐space renormalization‐group calculations. The interplay between the quenched randomness of the anisotropy and the annealed disorder introduced by the spin‐1 model leads to a rich phase diagram with a variety of phase transitions and reentrant behavior. Our results may be relevant to the study of the phase separation of3He–4He mixtures in porous media in the vicinity of the superfluid transition.