Is every formally real field isomorphic to a subfield of the reals?

Question

A formally real field is a field $K$ such that $-1$ is not a sum of squares in $K$. Clearly subfields of $\mathbb{R}$ are formally real. I also know finite fields and algebraically closed fields are never formally real. Then the next example I can think of is a field of rational functions of a formally real field, for example $\mathbb{Q}(t)$. But then mapping $t$ to some transcendental number in $\mathbb{R}$, e.g. $\pi$, I get an isomorphism to a subfield of the reals, in this case $\mathbb{Q}(\pi)$. Is there any example where I cannot get an isomorphism into a subfield of $\mathbb{R}$? Is this possible in division rings/ skew-fields?