When is the Kleisli category equivalent to the Eilenberg-Moore?

I think Beck's monadicity theorem provides an answer, it gives sufficient conditions for a functor $G: D \to C$ to be 'monadic' (i.e. has a left adjoint $F$ such that $D$ is the space of $GF$-algebras).

Applying this theorem to the forgetful functor $U : C_T \to C$ of the Kleisli, quite a few conditions are trivially satisfied, resulting in the following necessary and sufficient condition for the embedding $C_T \hookrightarrow C^T$ to be essentially surjective

$C_T$ has coequalizers of $U$-split parallel pairs and $U$ preserves those coequalizers

This condition is extremely technical, but if I understand the terms correctly a pair $f,g : A \to B$ has a split co-equaliser if there exists an arrow $h : B \to C$ such that the following commutes

$$ A \mathop{\rightrightarrows}^{f}_g B \mathop{\rightarrow}^h C $$ and there is a section $s$ of $h$ and a section $t$ of $f$ such that $g \circ t = s \circ h$.

Such a pair $f$, $g$ is called $U$-split if the above is true of $Uf$ and $Ug$.