A question of van den Dries and a theorem of Lipshitz and Robinson; not everything is standard

We use a new construction of an o-minimal structure, due to Lipshitz and Robinson, to answer a question of van den Dries regarding the relationship between arbitrary o-minimal expansions of real closed fields and structures over the real numbers. We write a first order sentence which is true in the Lipshitz-Robinson structure but fails in any possible interpretation over the field of real numbers. An o-minimal structure is by definition an expansion M of a linear ordering, such that every definable subset of the linear ordering is a finite union of intervals whose end points are in M∪ {±∞}. Although o-minimal expansions of discrete linear orderings do exist (e.g. 〈Z, <, z 7→ z + 1〉), these were recognized early on to have a relatively poor structure and therefore, in the above definition, one often assumes that the linear ordering is dense without endpoints. As was shown in [5], an o-minimal structures is, at least locally, one of the following three possibilities: It is degenerate (basically an expansion of the linear ordering by unary functions), an (interval in an) ordered vector space over an ordered division ring, or an expansion of a real closed field. Since ordered vector spaces over noncommutative ordered division rings essentially cannot be further expanded while preserving o-minimality (see [5]), it is within the third possibility, of expansions of real closed fields, where new o-minimal structures can be found. Indeed, construction of new o-minimal structures is usually carried out in this context, and much work has been done in this direction in the past twenty years. However, until very recently, all of these new o-minimal structures were expansions of the field of real numbers 〈R, <,+, ·〉. Of course, we know from model theory that every such structure M over R gives rise to an o-minimal expansion M∗ of a nonstandard real closed field. Now let M1 be a structure with the same universe as M, and whose relations are some of those definable with parameters inM. M1 may not be elementarily equivalent to any structure over the real