It is known that it consistent with $\mathsf{ZF}$ that there are no disjoint stationary subsets of $\omega_1$.
Is the assertion "there exist a stationary costationary subset of $\omega_1$" equivalent to, or does it imply/is it implied by some weak form of choice or some other more "standard" statement over $\mathsf{ZF}$? I couldn't find anything similar in Rubin and Howard standard reference "consequences of the axiom of choice" (in fact unless my pdf reader is badly broken the word "stationary" is never used in the book, which was a big surprise to me!)