Solovay's Theorem on Partitions of Stationary Sets and Weak Choice Principles There is a weak choice principle called $DC_\lambda$ which holds in $L(V_{\lambda+1})$ under the assumption of a non-trivial elementary embedding $$j:L(V_{\lambda+1})\prec L(V_{\lambda+1})$$ and it is known that this choice principle is not sufficient to split the ordinals below $\lambda^+$ (a regular cardinal) which have cofinality $\omega$ into disjoint stationary sets.


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*Is there a choice principle $\Phi$, which, when augmented with $DC_\lambda$ and strictly weaker than full AC that suffices to prove Solovay's Theorem on Partition of Stationary Sets? A little more specifically, what is the minimal $\Phi$ such that $T=ZF + DC_\lambda + \Phi$ where $$T\vdash \text{All stationary subsets of }\lambda^+\text{ have a disjoint partition into stationary sets}?$$ 
One such candidate could be $Unif(V_{\lambda+1}\times V_{\lambda+1})$: given any $R\subseteq V_{\lambda +1}\times V_{\lambda +1}$ there exists some function $f\subset R$ with the same domain as $R$.
(This question is related to both A proposed axiom of Laver (updated) and Model of ZF + $\neg$C in which Solovay's Theorem on stationary sets fails? .)

*Are there other weak choice principles $\Phi$ that could be considered? 

*Are there (perhaps) some partition properties with infinite exponents that would prohibit Solovay's Theorem? 

EDIT: While I am interested in the more general question (which I believe Asaf Karagila has addressed in the comments and chat), I am really specifically interested in the context that Woodin's Axiom $I_0$ holds. Specifically, any assertion $\Phi$ I'm looking for can't imply that $[\lambda]^\omega$ is well-ordered (in conjunction with $DC_\lambda$). 
 A: Although it may not be what you are looking for, this paper (called Splitting stationary sets from weak forms of Choice) contains some results along the lines of your questions. Theorem 3.1 of the paper shows (in the simplest case to state) that, assuming DC, if $\lambda$ is a regular cardinal and there is a function that chooses a countable cofinal subset of each ordinal below $\lambda$ of countable cofinality, then the set of ordinals of countable cofinality below $\lambda$ can be split into $\lambda$ many stationary sets. Corollary 3.3 shows that there is no elementary embedding from $V$ into $V$ given a stronger assumption : DC plus the statement that for every ordinal $\lambda$ there is a wellorderable $\mathcal{A} \subseteq [\lambda]^{\aleph_{0}}$ such that every element of $[\lambda]^{\aleph_{0}}$ intersects a member of $\mathcal{A}$ infinitely. The stronger hypothesis is needed to get that $\kappa^{+}_{\omega}$ (the ordinal where the stationary set splitting occurs in the standard argument) is regular. I believe that there are weaker principles that suffice to get the regularity of $\kappa_{\omega}^{+}$ in some of Shelah's papers on Choice-less PCF theory (number 835, for instance). I hope this helps. 
