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Allowing <jats:italic>r<\/jats:italic> and <jats:italic>k<\/jats:italic> to vary, we obtain a 2-parameter family of spaces that grow larger when <jats:italic>r<\/jats:italic> increases or <jats:italic>k<\/jats:italic> decreases, called the <jats:italic>multicover bifiltration<\/jats:italic>. Motivated by the problem of computing the homology of this bifiltration, we introduce two closely related combinatorial bifiltrations, one polyhedral and the other simplicial, which are both topologically equivalent to the multicover bifiltration and far smaller than a \u010cech-based model considered in prior work of Sheehy. Our polyhedral construction is a bifiltration of the <jats:italic>rhomboid tiling<\/jats:italic> of Edelsbrunner and Osang, and can be efficiently computed using a variant of an algorithm given by these authors. Using an implementation for dimension 2 and\u00a03, we provide experimental results. 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