Relation between well-orderings of $\mathbb{R}$, and bases over $\mathbb{Q}$

The problem is generally open. However, recently Liuzhen Wu, Liang Yu, Ralf Schindler and Mariam Beriashvili posted a preprint in which they prove the consistency of the existence of a Hamel basis and $\Bbb R$ cannot be well-ordered. Specifically, they show there is such a basis in Cohen's first model.

This can be found on Ralf's homepage.

There are actually two papers: "ZF + there is a Hamel basis" doesn't give that every infinite set of reals has a ctble. subset, see https://ivv5hpp.uni-muenster.de/u/rds/hamel_basis.pdf , and "ZF + DC + there is a Hamel basis" doesn't give that there is a w.o. of R, see https://ivv5hpp.uni-muenster.de/u/rds/hamel_basis_2.pdf