Wild theories with o-minimal open core

Let T be a consistent o-minimal theory extending the theory of densely ordered groups and let T′ be a consistent theory. Then there is a complete theory T⁎ extending T such that T is an open core of T⁎, but every model of T⁎ interprets a model of T′. If T′ is NIP, T⁎ can be chosen to be NIP as well. From this we deduce the existence of an NIP expansion of the real field that has no distal expansion.