  This paper provides a full characterization for when the expansion of a
complete o-minimal theory by a unary predicate that picks out a divisible dense
and codense subgroup has a model companion. This result is motivated by
criteria and questions introduced in the recent works concerning the existence
of model companions, as well as preservation results for some neostability
properties when passing to the model companion. The focus of this paper is
establishing the companionability dividing line in the o-minimal setting
because this allows us to provide a full and geometric characterization.
Examples are included both in which the predicate is an additive subgroup, and
where it is a multiplicative subgroup. The paper concludes with a brief
discussion of neostability properties and examples that illustrate the lack of
preservation (from the "base" o-minimal theory to the model companion of the
expansion we define) for properties such as strong, NIP, and NTP$_2$, though
there are also examples for which some or all three of those properties hold.
