Constructing models over a predicate

Given a complete theory T and a definable set X, we can consider the class of structures X(M) (with the induced structure) for M a model of T. When is this itself an elementary class, i.e. when does every model of the theory of the induced structure extend to a model of T? This "existence" property has been studied in particular in the context of Gaifman's conjecture that it is implied by relative categoricity, i.e. uniqueness over X of the model when it exists. I will discuss some of the history of this, and some work with Itay Kaplan and Pierre Simon which shows that this existence property holds if X is stably embedded and T is simple (or just rosy), going via X being what we call "arid".