I have a braided monoidal, semisimple linear category $\mathcal{C}$. (Imagine representations of a semisimple quasitriangular Hopf algebra.) I also have a monad $(T,\mu,\eta)$ on it, however, $T$ is not necessarily monoidal. (Imagine left-tensoring with an algebra $A$ internal to $\mathcal{C}$, so $T = A \otimes -$.)

Is there a general way to decide whether the Kleisli category or the Eilenberg-Moore category are braided monoidal?

I already know from Wikipedia that if $T$ is lax monoidal and $\mu$ and $\eta$ are monoidal natural transformations, then the Kleisli category is monoidal as well, but what about the Eilenberg-Moore category? And do either have a braiding coming naturally from $\mathcal{C}$?

Finally, when are Kleisli and Eilenberg-Moore categories semisimple? For my example, I guess the Kleisli category is not semisimple, but the Eilenberg-Moore category is, but I'm still a bit confused.