Something I'm noticing is that many of these preorders involve some sort of function \\( f: \mathbb{C} \to \mathbb{R}\\). Then we use the familiar ordering from \\(\mathbb R\\) to give us an ordering on \\(\mathbb{C}\\) by saying that $$ z \leq z' \text{ iff } f(z) \leq f(z').$$

To make monoidal-ness of the ordering on \\(\mathbb C\\) follow from the monoidal-ness of \\(\mathbb R\\), we would need that \\( f(z + z') = f(z) + f(z')\\). In other words, \\(f\\) must be a homomorphism!

I suspect that we could play this trick with other monoidal pre-orders besides \\(\mathbb R\\).

To make monoidal-ness of the ordering on \\(\mathbb C\\) follow from the monoidal-ness of \\(\mathbb R\\), we would need that \\( f(z + z') = f(z) + f(z')\\). In other words, \\(f\\) must be a homomorphism!

I suspect that we could play this trick with other monoidal pre-orders besides \\(\mathbb R\\).