Working in the language of ordered rings, which we take to have type $(+ - \times < 0\, 1)$, can anyone give an example of a discrete ordering on the polynomial ring in two variables $\mathbb{Z}[x,y]$ such that the resulting ordered ring does not satisfy the universal theory of the integers? 

It is not difficult to show that as a ring, i.e. forgetting the less-than symbol, the ring $\mathbb{Z}[x,y]$ does in fact satisfy the universal theory of the ring of integers. The problem is to find a discrete ordering on $\mathbb{Z}[x,y]$ such that some system of inequalities is solvable in $\mathbb{Z}[x,y]$ but not in the integers, or, on the contrary, to prove that there is no such ordering.