Does there exist an ordered field where only addition preserves positivity?

Question

I am trying to find a field (I'll settle for other stuff - some type of ring) that is ordered and $a>0,\,b>0$ implies $a+b > 0$ yet $a,b > 0$ doesn't imply $ab > 0$.

Much of the problem seems to be working to satisfy the distributive law.

Cheers.

edit: and $a>0$ or $a=0$ or $a<0$ holds for all $a$ and only one relation does in-fact hold.

Answer 1

Take $\mathbb{C}$ with lexicographic order; that is, $a + bi \le c + di$ if $a \le c$ or $a = c$ and $b \le d$.

Answer 2

Another example: Reverse the usual order on $\mathbb R$ (or any other ordered field).