The generalized method of cells is a micromechanics model that is generally quite accurate at the macrolevel but not always accurate at the microlevel. This is due to the absence of so-called shear coupling, which provides the required bridge between macroscopically applied normal (shear) stresses and the microscopic shear (normal) stresses necessary for an accurate estimate of microlevel quantities. To overcome this deficiency, a new micromechanics model has been developed for the response of multiphase materials with arbitrary periodic microstructures, named high-fidelity generalized method of cells in part because it employs the same microstructural discretization as the original generalized method of cells. The model's framework is based on the homogenization theory, but the method of solution for the local fields borrows concepts previously employed in constructing the higher-order theory for functionally graded materials, in contrast with the typical finite element-based solution strategies. The model generates the average stress-strain response of heterogeneous materials such as ceramic, metal, and polymeric matrix composites, as well as the internal or microlevel stress and strain fields, with excellent accuracy. The model is employed to investigate the response of a metal matrix composite with locally irregular, but periodic, fiber distributions, and it is shown that irregular architectures affect shear and normal stress-strain response in a different manner. The new model's ability to capture these differences is attributed to the shear-coupling effect absent in the original model.