@Jonathan – let's take a look at your minimal-poset-that-isn't-a-semilattice above.

suppose it has a monoidal structure that respects the order, with \$$d\$$ as the unit \$$I\$$.

then \$$a\otimes b \ge a\otimes d = a\otimes I = a\$$

and \$$a\otimes b \ge d\otimes b = I\otimes b = b\$$

this is a contradiction, since \$$a\$$ and \$$b\$$ do not have a common upper bound

similar arguments show the unit can't be \$$a\$$, \$$b\$$ or \$$c\$$ either.

so there is no monoidal structure that respects the ordering – QED