Via Cantor's back-and-forth method we know that the linearly ordered set of all rational numbers and the linearly ordered set of all real algebraic numbers are isomorphic.
But from the point of view of what people usually do with rational numbers or with algebraic numbers, the order isomorphisms yielded by Cantor's proof are icky.
Are there any that are well behaved and have nice mathematical properties?
I'm not sure if this should be an answer, since it's not a proof but more heuristic evidence for a "no". This question is very similar to Order preserving bijection from $\mathbb Q\times \mathbb Q$ to $\mathbb Q$, but the difference is that the order on the real algebraic numbers is quite far removed from the means by which we enumerate them (in terms of coefficients and "root number"). Indeed I think that the problem would be solved if there was a nice closed form expression for the order predicate given a coding $\langle a_0,\dots,a_n,i\rangle$ of an algebraic number (the $i$-th root of $a_0+\dots+a_nx^n=0$). Even this coding is problematic because it doesn't eliminate double roots or imaginary roots, and "root number" is usually determined by sorting on real part, which is assuming the consequent in this case. So I think that you can't answer this question without making a breakthrough first in the representation and evaluation of polynomial roots.