Strictly associative coproducts

I'm going to cheat and assume the axiom of global choice. We well-order the universe of sets, so that every set $A$ is equipped with a well-ordering (of type $\alpha_A$). We then set $A \amalg B$ to be the von Neumann ordinal of $\alpha_A + \alpha_B$, and set $A \hookrightarrow A \amalg B$ and $B \hookrightarrow A \amalg B$ to be the obvious order-preserving maps. It is then clear that $A \amalg (B \amalg C) = (A \amalg B) \amalg C$ as sets (because ordinal addition is strictly associative) and they are even isomorphic as coproducts (because the insertions are the same).

Obviously, this fails to be commutative...