  Let $T$ be an o-minimal theory extending the theory of real closed ordered
fields. An $H_T$-field is a model $K$ of $T$ equipped with a $T$-derivation
such that the underlying ordered differential field of $K$ is an $H$-field. We
study $H_T$-fields and their extensions. Our main result is that if $T$ is
power bounded, then every $H_T$-field $K$ has either exactly one or exactly two
minimal Liouville closed $H_T$-field extensions up to $K$-isomorphism. The
assumption of power boundedness can be relaxed to allow for certain exponential
cases, such as $T = \operatorname{Th}(\mathbb{R}_{\operatorname{an},\exp})$.
