Is there some theory related to this...
Let $(\{X_\alpha\}_\alpha,f_{\alpha\beta}:X_\beta\rightarrow X_\alpha\}_{\alpha\preceq\beta})$ be an inverse directed system of non-empty sets resp. surjective functions. Under which circumstances (axiom of choice assumed) do we have a choice $\{x_\alpha\}_\alpha$ of elements in $X_\alpha$ which is consistent in the sense of $f_{\alpha\beta}(x_\beta)=x_\alpha$? Or are there some prominent counterexamples?
OK, I think I came up with a counterexample at least. (I didn't actually think that the assumption would hold but it still took me some time to confirm this.)
Let our directed set be $\aleph_1$ and $X_\beta$ consisting of all finite sequences of ordinals $\beta_0<\beta_1<\cdots<\beta_{n-1}\le\beta<\beta_n<\aleph_1$. Then let $f_{\alpha\beta}(\beta_0,\ldots,\beta_n)$ equal the unique prefix $(\beta_0,\ldots,\beta_k)$ that is contained in $X_\alpha$. Clearly this forms an inverse directed system and also the $f_{\alpha\beta}$ are surjective. But assuming there is some consistent choice $s_\alpha\in X_\alpha$, we can see that the $s_\alpha$ get arbitrary long, which contradicts regularity of $\aleph_1$ and the fact that the $f_{\alpha\beta}$ do not increase the length of sequences.