Fields that can be ordered in more than one way

Consider $F=\mathbb{Q}(x)$. You can order this field in uncountably many different ways. For instance, for every transcendental $\alpha\in\mathbb{R}$, the isomorphism $F\to\mathbb{Q}(\alpha)\subset\mathbb{R}$ sending $x$ to $\alpha$ induces an ordering of $F$, and this ordering is different for each $\alpha$. Since $F$ has only countably many automorphisms (an automorphism is determined by where it sends $x$), these give uncountably many orderings that are not related by automorphisms.

Here's a smaller example: let $f$ be an irreducible quartic polynomial with four real roots. If $\alpha$ is any root of $f$, then $\mathbb{Q}(\alpha)$ can be ordered in four different ways, corresponding to the four real embeddings of $\alpha$

However, most such polynomials $f$ will have Galois group $A_4$, and $\mathbb{Q}(\alpha)$ will have no automorphisms at all.