We advance a three-dimensional phenomenological model for the magneto-mechanical behavior of magnetic shape memory alloys. Moving from micromagnetic considerations, we propose a thermodynamically consistent constitutive relation which is able to reproduce the magnetically-induced martensitic transformation in single crystals. Existence results for the constitutive relation problem as well as for the corresponding quasi-static evolution system are illustrated and convergence of time- and space–time-discretizations are recorded. Eventually, we present algorithmic considerations and we numerically test the model in order to assess its ability in reproducing the typical response of magnetic shape memory alloys, as well as in recovering standard shape-memory and pseudo-elastic behaviors when no magnetic field is present.