Bilinear factorization of algebras

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 factorization 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.