Let $M$ be a length space, and let $\Pi$ be a finite set of geodesics in this space that are the unique geodesic between their endpoints. Additionally, suppose that the intersection of any two distinct $\pi_1, \pi_2 \in \Pi$ contains at most a single point.
Is it then necessarily possible to choose one of the two possible directions for each $\pi \in \Pi$ such that the resulting system of paths is acyclic? By "acyclic," we mean that there is no closed path in $\bigcup \limits_{\pi \in \Pi} \pi$ that only travels any given path $\pi$ in its chosen direction.
I somewhat suspect that this isn't possible, but I've been unable to think of a counterexample. It is true for many natural length spaces, e.g. $R^n$.