Proposition: Double-ordering geometry

Let $A={\mathrm{supp}}_{\mathrm{fun}}F$ and $B={\mathrm{supp}}_{\mathrm{fun}}G$. Both $F\gtrsim G$ and $G\gtrsim F$ hold exactly when no point of $A$ is causally related in either direction to a point of $B$. In particular, $A\cap B=\varnothing$.

Proof. The first condition excludes pairs $a\in A,b\in B$ with $a\in J^{-}(b)$. The second excludes pairs with $b\in J^{-}(a)$. These are precisely the two directions of causal relation. If $a=b$, reflexivity gives both relations, so coincidence is excluded as well.