Monads and modules, and the bicompletion under Kleisli and Eilenberg–Moore objects

Question

In The Formal Theory of Monads, Street proves that a 2-category $\mathscr C$ admits the construction of algebras when the inclusion $\mathscr C \to \mathbf{Mnd}(\mathscr C)$ has a right adjoint. In The Formal Theory of Monads II, Street and Lack show the same property holds when the inclusion $\mathscr C \to \mathbf{EM}(\mathscr C)$ has a right adjoint, where $\mathbf{EM}$ is the completion under Eilenberg–Moore objects. A dual situation holds for Kleisli objects, where we instead require a left adjoint to $\mathscr C \to \mathbf{Mnd}(\mathscr C^{\mathrm{op}})^{\mathrm{op}}$ or to $\mathscr C \to \mathbf{KL}(\mathscr C)$.

We can to some extent combine the constructions of $\mathbf{Mnd}$ and $\mathbf{Mnd}({-}^{\mathrm{op}})^{\mathrm{op}}$, by taking the 2-category $\mathbf{Mod}(\mathscr C)$ of monads and modules in $\mathscr C$ (assuming $\mathscr C$ has local coequalisers). The 2-category $\mathscr C$ has Kleisli objects when $\mathscr C \to \mathbf{Mod}(\mathscr C)$ has a left adjoint, and has Eilenberg–Moore objects when it has a right adjoint (e.g. see Lemma 43 of Wood's Proarrows II).

1. Is there a construction with the same relation to $\mathbf{Mod}$ that $\mathbf{EM}$ has to $\mathbf{Mnd}$? I expect it might be given by a bicompletion under Kleisli and Eilenberg–Moore objects. Does this exist in the literature, or may it be derived by general techniques if not?
2. To elaborate on "with the same relation": what is the precise relationship between $\mathbf{EM}$ and $\mathbf{Mnd}$? It is shown in The Formal Theory of Monads II that the constructions agree on objects and 1-cells, but they do not express any further relation between the two. However, both have the same property in terms of bestowing their underlying category with Eilenberg–Moore objects in the presence of a right adjoint to inclusion, which suggests a stronger connection.

While these two questions are distinct, I feel they are closely related enough to be worth grouping; I can split them if an answer to one does not lead to an answer to the other.

Answer 1

An answer to the second question has been provided in the recent preprint What is the universal property of the 2-category of monads? by Lack–Miranda. The 2-monad morphism $\mathbf{Mnd}(\mathscr C) \to \mathbf{EM}(\mathscr C)$ is the (bijective-on-objects functors)-enriched completion of the identity 2-functor on $\mathscr C$.

It may be that this suggests an approach for answering (or making sense of) the first question.