We study the anti-plane elasticity problem of an $N$ -phase composite comprising of an internal anisotropic elastic elliptical inhomogeneity bonded to an infinite anisotropic elastic matrix through $N-2$ intermediate anisotropic elastic coatings when the matrix is subjected to uniform remote anti-plane shear stresses. We prove that the internal stress field within the elliptical inhomogeneity is still uniform when the geometries of the outer $N-2$ elliptical interfaces are designed for given elastic constants for the $N-2$ coatings, given aspect ratio of the innermost elliptical interface (with its two principal axes along the two coordinate axes) and given thickness parameters of the coatings. All the 2 $N$ complex coefficients appearing in the one-to-one mapping functions can be simply determined using the derived recurrence relation. The internal uniform stress field within the elliptical inhomogeneity is obtained using the transfer matrix method. The general result is demonstrated through an example of a double coated elliptical inhomogeneity.
