Is the well-orderability of the reals independent from dependent choice?

Question

Working in $ZF$+"There is a well-order of the real numbers", is there any known model that also satisfies $\neg DC$? I have tried to look at this in Jech's "The Axiom of Choice", but I'm not very familiar with symmetric extensions or permutation models, so I don't know whether the techniques used to break $DC$ would work if the reals are well-ordered. In principle I don't see why not, but I've learned to be cautious about the $ZF$ world.