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There are some models in which $2^\omega$ is not wellorderable but there is a free ultrafilter over $\omega$. What about the consistency of: $2^\omega$ is not wellorderable + AC for countable sets of reals + there is a free ultrafilter over $\omega$ with a wellorderable base? There is one such model $N$. Namely let $M$ be the $\omega_2$-long or $\omega_1$-long countable-support iterated Sacks extension of $L$, and let $N$ consist of all sets hereditarily definable in $M$ from an $\omega$-sequence of ordinals. But this is a rather peculiar model. I wonder is there anything essentially simpler and possibly better known.

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