On any spacetime $(M,g)$ we can form the causal ordering $\leq$, where for any two points $a,b \in M$ we have that $a \leq b$ iff there exists some future directed, non-spacelike curve from $a$ to $b$. This causal ordering is transitive, irreflexive, antisymmetry and dense in general. However, causal oddities like CTC's manifest as violations of irreflexivity and antisymmetry.
I'm interested in the directedness of the causal order (unfortunate name, I know). Recall that a relation $\leq$ is directed iff for any two elements $a,b \in M$ there is some third element $c \in M$ such that both $a \leq c$ and $b \leq c$.
I can see that a Minkowksi space has a directed causal order. I have no proof, but the picture below is good enough to convince me. Clearly anything in the shaded area will serve as the witness for the condition. Note also that the condition is trivially met for timelike or lightlike-related points.
I have heard that for non-flat spacetimes, directedness of the causal order is not guaranteed. Are there any concrete examples of this? And, moreover, is there some appropriate causality condition one can impose on the spacetime in order to guarantee directedness of the causal order?