  Building on coprincipal mesoprimary decomposition [Kahle and Miller, 2014],
we combinatorially construct an irreducible decomposition of any given binomial
ideal. In a parallel manner, for congruences in commutative monoids we
construct decompositions that are direct combinatorial analogues of binomial
irreducible decompositions, and for binomial ideals we construct decompositions
into ideals that are as irreducible as possible while remaining binomial. We
provide an example of a binomial ideal that is not an intersection of binomial
irreducible ideals, thus answering a question of Eisenbud and Sturmfels [1996].
