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.
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2You may apply the Jech-Sochor theorem to find a model: apply this theorem to a permutation model of $\mathsf{ZF+\lnot DC}$, and correspond atoms in the permutation model to sets of very high rank (possibly greater than $\omega+4$ would be fine.) It could not be hard to find a direct construction, but I bet Asaf will answer it. – Hanul Jeon Jul 16 '21 at 13:50