In this study, the effective constitutive behavior of heterogeneous materials comprising bimodular phases is investigated through a computational approach and by referring to Curnier-type bimodularity. Different microstructures characterized by spherical inclusions or voids are addressed, by analyzing different loading scenarios. Numerical results, obtained via an iterative finite-element scheme, highlight the influence of intraphase constitutive non-linearities induced by the tension/compression transition of the local material stiffness. Moreover, coupling effects between hydrostatic and deviatoric states are elucidated. The macroscale material response results dependent on the loading condition, and it is driven by perturbative effects of heterogeneous fields locally induced by pores or inclusions.