Many engineering materials, such as semi-crystalline polymers, possess heterogeneous, hierarchical, and complex morphologies and require accurate predictions of their macroscopic behavior from underlying microscale geometry and mechanisms. Computational homogenization offers a robust framework for addressing this challenge, bridging scales through nested boundary value problems (BVPs) solved at both macroscopic and microscopic levels. Blanco and coworkers\cite{ref1} established a complete variational formulation for a multi-scale constitutive theory of solids based on computational homogenization. It rests on the rigorous definition of the space of admissible microscopic displacements, coupled with the Principle of Multi-Scale Virtual Power (PMVP). This work focuses on mean-field homogenization techniques, which simplify the interaction between material phases based on the behavior of each phase in an auxiliary single inclusion problem. These methods provide computational efficiency and acceptable macroscopic accuracy\cite{ref2}, and we seek to integrate them into the Principle of Multi-Scale Virtual Power (PMVP) and its dual, the Principle of Multi-Scale Complementary Virtual Power (PMCVP). We focus on a rigorous definition of the minimal spaces of kinematically and statically admissible fluctuation fields, allowing for incompatible displacements and non-equilibrium stresses. We then consider the different models obtained from more stringent restrictions on this minimal space. These formulations result in homogenization rules that differ from standard volume averages and include models coinciding with the $\tilde{\boldsymbol S}$- and $\tilde{\boldsymbol D}$-inclusion models proposed by\cite{ref3} for large-strain semi-crystalline polymer behavior, as well as the Taylor and Sachs models, which coincide with the minimal and most constrained spaces of admissible fluctuation fields. Their computational efficiency and accuracy are confirmed through numerical examples, which consider different geometries such as ellipsoids and laminates and elastic, viscoelastic, and viscoplastic constitutive behaviors. A practical example concerning the simulation of large-strain plastic deformation and texture evolution in high-density polyethylene is also presented\cite{ref3}.