This paper uses an efficient surface integral equation-based MoM (SIE-MoM) scheme to study the electromagnetic shielding properties of arbitrary-shape multilayered anisotropic composite enclosures. In this scheme, each anisotropic composite layer is modeled by an equivalent homogeneous anisotropic medium with tensorial characteristics. The method treats each layer, separately, using the surface equivalence theorem. After applying the suitable boundary conditions, the SIEs solved by the Galerkin’s-MoM. This computation includes, the expansion of equivalent surface current densities by proper basis functions, the rotation of dyadic Green’s functions of the infinite space filled with the equivalent anisotropic material used for each layer, and finally the inner product of both sides of integral equations to express the SIEs in the form of matrix equations solved through an efficient inversion process of a sparse block-tridiagonal impedance matrix. The rotation angle of dyadic Green’s functions is the angular deviation between the specified global coordinate system and the local principal one for anisotropic materials, which is determined by diagonalizing the permittivity tensor of the equivalent anisotropic layer defined in the global coordinate system. The validity and efficiency of the presented method are demonstrated for a three-layer irregular-shaped anisotropic composite enclosure by comparing the obtained results with those obtained using FE-solver of the commercial CST studio. Finally, a set of sensitivity analysis is done to examine the various parameters that affect the shielding properties of a two-layer cubic anisotropic composite enclosure.