For the purposes of this question, totally ordered fields are ordered fields in which every element is comparable under the ordering: $x \geq y$ or $y \geq x$. Thus in this terminology, in an ordered field, sometimes called a partially ordered field, two elements may be order-incomparable.
Bourbaki Algebra II A.VI.20 restricts attention to totally ordered fields on the grounds that ordered fields that are not totally ordered are "very pathological.'' In explanation of this, they refer to an exercise that says that for a lattice ordered field, being totally ordered is equivalent to each of a number of natural properties, such as that $x^2 \geq 0$ for all $x$, or that $x>0$ implies $x^{-1} >0$.
My question is whether there is much work discussing non-lattice ordered fields that are not required to be totally ordered but which nevertheless have good properties. In particular, are there any references that provide an introduction?