Another ordering on the set of real numbers

Question

Is there a total order relation on the set of real numbers that is compatible with the field structure but does not satisfy the completeness for the associated order topology?

Answer 1

No. The ordering of $\mathbb{R}$ compatible with the field structure is unique.

In fact, assume we have another ordering $\succ$ such that $0\succ a$ for some $a$ that is positive in the standard order. Then we have $a=b^2$ for some $b$, which means $0\succ b^2$ but either $b$ or $-b$ is positive with respect to $\succ$, so we have $b^2=(-b)^2\succ 0$. This is a contradiction.