There are expansions of dense linear orders by open sets (of arbitrary arities) such that all of the following hold:

1) Every definable set is a boolean combination of existentially definable sets.
2) Some definable sets are not existentially definable.
3) Some projections of closed bounded definable sets are somewhere both dense and codense.
4) There is a unique maximal reduct having the property that every unary definable set either has interior or is nowhere dense. It properly expands the underlying order, yet is still rather trivial.

At least some of these structures come up naturally in model theory. For example, if G is a generic predicate for the real field, then the expansion of (G,<) by the G-traces of all semialgebraic open sets is such a structure, which moreover is interdefinable with the structure induced on G in (R,+,x,G).