This article is concerned with the modeling of the magnetic shape memory alloy (MSMA) constitutive response caused by the reorientation of martensitic variants under mechanical and magnetic fields. The presented model is able to better capture the complexity of the magnetomechanical MSMA behavior by accounting not only for the mechanism of field-induced variant reorientation, but also the magnetization rotation away from magnetic easy axes and the magnetic domain wall motion at low stress and magnetic field levels. Following the general formulation of the model, reduced versions of the constitutive equations are derived for three specific loading cases: (1) magnetic-field-induced variant reorientation at constant stress; (2) stress-induced variant reorientation at constant magnetic field; (3) variant reorientation under collinear magnetic field and stress with perpendicular bias field. For each of these cases the nonlinear and hysteretic strain and magnetization response of MSMAs are predicted and compared to experimental data where available. The relation of critical stresses and magnetic fields for the activation of the reorientation process are visualized in a variant reorientation diagram. The captured loading-history-dependent macroscopic material response is explained in detail by connecting it to the evolution of the crystallographic and magnetic microstructure as represented by a set of internal state variables.