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\$$.