The overall behavior of a two-dimensional lattice of voids embedded in an anisotropic elastic matrix is investigated in the limit of vanishing porosity f. An effective-medium model (of the Clausius-Mossoti type), which accounts for elastic interactions between neighboring voids, is compared to fast Fourier transform numerical solutions and, in the limits of infinite anisotropy, to exact results. A crossover between regular and singular dilute regimes is found, driven by a characteristic length which depends on f and on the anisotropy strength. The singular regime, where the leading dilute correction to the elastic moduli is an O(f1/2), is related to strain localization and to change in character—from elliptic to hyperbolic—of the governing equations.