This paper addresses the complexity reduction of stochastic homogenization of a class of random materials for a stationary diffusion equation. A cost-efficient approximation of the correctors is obtained using a method designed to exploit quasi-periodicity. Accuracy and cost reduction are investigated for local perturbations or small transformations of periodic materials as well as for materials with no periodicity but a mesoscopic structure, for which the limitations of the method are shown. Finally, for materials outside the scope of this method, we propose to use the approximation of homogenized quantities as control variates for variance reduction of a more accurate and costly Monte Carlo estimator (using a multifidelity Monte Carlo method). The resulting cost reduction is illustrated in a numerical experiment and compared with a control variate method from weakly stochastic homogenization. The limits of this variance reduction technique are tested on materials without periodicity or mesoscopic structure.