TITLE: Is monadicity preserved by the underlying functor?
QUESTION [3 upvotes]: Let $\mathcal{V}$ be a monoidal closed (complete, cocomplete, reasonable...) category.
Let $\mathsf{T}$ be an enriched monad over $\mathcal{V}$. The forgetful functor $\mathsf{U}: \mathsf{Alg}(\mathsf{T}) \to \mathcal{V}$ is tautologically monadic in  $\mathcal{V}$-Cat. If we pass to the underlying categories $\mathsf{U}_0: \mathsf{Alg}(\mathsf{T})_0 \to \mathcal{V}_0$, do we still get a monadic functor?
$\mathsf{U}_0$ is still a right adjoint because $(-)_{0}: \mathcal{V}\text{-Cat} \to \text{Cat}$ is a $2$-functor, but
I expect a negative answer to my question. Yet, I can't find a counterexample.

REPLY [0 votes]: The answer of Ben MacAdam is very solid and concrete, but I could not believe his result because it did not meet my intuition. In the process of trying to prove him wrong, I managed to have a better understanding of the situation. Indeed Ben is just right.
Excursus. For a monad $\mathsf{T}$ in $\mathcal{V}$-Cat, its category of algebras $\mathsf{Alg}(\mathsf{T})$ is a lax limit. Thus Ben's result shows that the $2$-functor $(-)_0$ preserve a certain kind of $2$-limit. From this point of view, his result is even more ambitious, and I could not believe it. Until...
Yet another proof. When $(\mathcal{V}, \otimes, I)$ is cocomplete, there is an adjunction $ (-)^I:\text{Set} \leftrightarrows \mathcal{V}: \mathcal{V}(I,-) $. Such an adjunction yields a left $(2$-)adjoint for $(-)_0$, $$F: \text{Cat} \leftrightarrows \mathcal{V}\text{-Cat} : (-)_0. $$
This means in particular that $(-)_0$ preserves $2$-limits. As shown by Gray in
The existence and construction of Lax limits, a lax limit is always the $2$-limit of another diagram (and this construction changes the diagram functorially). This proves that $(-)_0$ must preserve Eilenberg-Moore objects.
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