In what classes of fields does CSB hold? That is to say, in what classes of fields is it true that if there exist embeddings $F\to K$ and $K\to F$ then $F$ and $K$ must be isomorphic?

I know this holds for algebraically closed fields, but all of the counter-examples I've seen are variations on the same idea ($F=\overline{\mathbb Q(x_0,x_1,x_2,...)}$ and $F(x)$).

Does CSB hold for fields of finite transcendence degree? What about for fields with no algebraically closed subfields?