The mechanics of laminates made up of elastic alternating stiff/soft layers is dominated by bending and membrane actions in the stiff layers, while transverse shear deformations concentrate in the soft interlayers producing significant zigzag effects [1]. In plates additionally, curved geometry and/or large deformations can induce an important thickness strain in the interlayers. This work presents a rotation-free Total Lagrangian large deformation curved shell formulation to capture reliably and accurately with a minimal number of DOFs such a structural response. Each stiff layer is modelled as a Kirchoff-Love shell [2]. Without additional DOFs, the generic soft interlayer is described by a solid-shell model [1] expressing, in finite kinematics, the displacement of top and bottom interfaces as a function of the mid-surface displacement of the stiff layers it couples. The model is characterized by a very sparse stiffness matrix compared to high order theories and needs much fewer DOFs than 3D and layer-wise models without missing any relevant information. A NURBS discretization meets the high continuity of the weak form with a patch-wise reduced quadrature to avoid locking enhancing also the efficiency. All integrals along the thickness are pre-computed just once. Finally, the small membrane strains assumption [2] is exploited to simplify the strain differentiation.