What's a non-standard model of Tarskian Euclidean geometry?

Question

Tarski's axioms (see here: http://en.wikipedia.org/wiki/Tarski%27s_axioms) are a first-order axiomatization of Euclidean Geometry. Now, I believe the standard model for the axioms is the real number plane (for 2D plane geometry) and 3D space (for the 3D form) with "congruence of segments" and "betweenness" defined in the usual ways via the distance formula. But what does a non-standard model of the axioms look like? I'm especially curious about how the "continuity axiom (schema)" constrains the choice.

Answer 1

I'm going to answer my own question! I found this paper, which gives the answer. It's called "Old and New Results in the Foundations of Elementary Plane Euclidean and Non-Euclidean Geometries" by MJ Greenberg, 2010.

http://www.maa.org/programs/maa-awards/writing-awards/old-and-new-results-in-the-foundations-of-elementary-plane-euclidean-and-non-euclidean-geometries

From the paper:

Geometrically, Tarski-elementary plane geometry certainly seems mysterious, but the representation theorem illuminates the analytic geometry underlying it: its models are all Cartesian planes coordinatized by real-closed fields.

(pg. 215) (pg. 18 of the paper)

So Qiaochu (in the comments to this question) was right, apparently.