The manuscript presents a refined two-dimensional (2D) formulation, based on higher-order theories, for the coupled magneto-electro-elastic analysis of laminated, doubly-curved shells. The geometry of the structure is described using curvilinear principal coordinates, and the unified formulation is adopted to represent the variation of the unknown field variables along the thickness direction. Based on the Equivalent Layer-Wise (ELW) methodology, the model allows for the enforcement of displacement fields as well as electric and magnetostatic potentials at the top and bottom surfaces. The constitutive relations account for both piezoelectric and piezomagnetic coefficients and enable a full coupling between the electric and magnetic fields and flux components through magneto-electric coefficients. The governing equations are derived under quasistatic conditions from the Master Balance principle by evaluating the stationary configuration of the total energy of the system. A semi-analytical solution is, then, obtained using the Navier approach. In the post-processing, an efficient recovery procedure is adopted to reconstruct the through-the-thickness distributions of both primary and secondary variables. A systematic comparison with numerical predictions obtained from three-dimensional (3D) commercial finite element software is carried out through some numerical examples. In this way, the results highlight both the accuracy and the computational efficiency of the proposed methodology for multifield analysis.