Green and Wald have presented a mathematically rigorous framework to study, within general relativity, the effect of small-scale inhomogeneities on the global structure of space-time. The framework relies on the existence of a one-parameter family of metrics that approaches the effective background metric in a certain way. Although it is not necessary to know this family in an exact form to predict properties of the backreaction effect, it would be instructive to find explicit examples. In this paper, we provide the first example of such a family of exact nonvacuum solutions to the Einstein equations. It belongs to the Wainwright-Marshman class and satisfies all of the assumptions of the Green-Wald framework.