We study the (so-called bilinear) factorization problem answered by a weak wreath product (of monads and, more specifically, of algebras over a commutative ring) in the works by Street and by Caenepeel and De Groot. A bilinear factorization of a monad R turns out to be given by monad morphisms A → R ← B inducing a split epimorphism of B-A bimodules B ⊗ A → R. We prove a biequivalence between the bicategory of weak distributive laws and an appropriately defined bicategory of bilinear

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... n structures. As an illustration of the theory, we collect some examples of algebras over commutative rings which admit a bilinear factorization; i.e. which arise as weak wreath products.