 Solution

Yes. The pure order on the reals is a dense linear order without endpoints and has quantifier elimination for dense linear orders without endpoints. Over a finite parameter set $a_1<\cdots <a_m$, a complete one-variable type either specifies equality to one of these points or specifies one of the open intervals and exterior rays they determine.

Each such interval or ray contains a real number. Hence every complete one-variable type over a finite parameter set is realized. The successive-coordinate characterization gives saturation for all finite tuples. This uses finite parameter sets; it does not say that arbitrary types over a countably infinite set of parameters are realized.