Well-ordering of power set of $\omega$ Assuming initially the background set theory ZF, what is the exact status of the existence of a well-ordering on the power set of $\omega$? How much needs to be added to guarantee this?
 A: More or less the only you need to require it explicitly.
It is consistent with any amount of Dependent Choice that the reals cannot be well ordered. Just blow the continuum to be large enough, and preserve the amount of required Dependent Choice.
I am not sure, however that this appears explicitly anywhere in print.
The sketch of the proof is as follows:

Start with a model of $\sf ZFC+GCH$, e.g. $V=L$, fix a regular $\kappa$, and consider the forcing $\Bbb P=\operatorname{Add}(\omega,\kappa^+)$. For a permutation $\pi$ of $\kappa^+$, $\pi$ acts naturally on the conditions of $\Bbb P$ as follows $$\pi p(\pi\alpha,n)=p(\alpha,n),$$ taking the normal filter of subgroups to be generated by groups which are the identity on some set of size $\leq\kappa$.
The fact that the reals cannot be well-ordered is proved via standard symmetric arguments. To see that $\sf DC_\kappa$ holds as well, note that any name for a sequence of hereditarily symmetric names of length $\kappa$ must be symmetric as well, simply due to chain conditions and the fact that $\kappa^+$ is regular of cofinality $>\kappa$.
Therefore the symmetric extension is closed under $\kappa$-sequences in the full (choice-y) extension, and it is not hard to show this implies $\sf DC_\kappa$ holds in the symmetric extension; and as we remarked, the reals cannot be well-ordered there.

Boolean Prime Ideal theorem holds in Cohen's first model, so certainly it does not prove a well ordering of the continuum.
