Can we find a non-cartesian monoidal category where for each pair of objects the tensor product $A \otimes B$ is a cartesian product of $A$ and $B$?

Let me explain.

A monoidal category $(\mathsf{C}, \otimes, I, a, r, \ell)$ is a category $\mathsf{C}$ with a tensor product $\otimes$, a unit object $I$, an associator $a$, and left and right unitors $r$ and $\ell$ obeying the pentagon and triangle identities. A cartesian category is one with binary products and a terminal object.

Given a cartesian category $\mathsf{C}$, and choosing a terminal object, and choosing a product cone

$$ A \leftarrow A \times B \rightarrow B$$

for each pair of objects $A,B$, there is a well-known way to make $\mathsf{C}$ into a monoidal category. To create this monoidal category, first we take $A \otimes B$ to be the object $A \times B$ in the chosen product cone for $A$ and $B$. Then we define $\otimes$ on morphisms using the universal property of the chosen product cones. Similarly, we define the associator

$$ a_{A,B,C} \colon (A \otimes B) \otimes C \to A \otimes (B \otimes C)$$

using the universal property of the chosen product cones. Then we take the unit object $I$ to be the chosen terminal object, and define the left and right unitors

$$ \ell_A \colon I \otimes A \to A, \qquad r_A \colon A \otimes I \to A$$

using the universal property of the chosen product cones and the fact that $I$ is terminal. We obtain a monoidal category $(\mathsf{C}, \otimes, I, a, r, \ell)$ called a 'cartesian monoidal category'.

More generally, any monoidal category equivalent as a monoidal category to one of this form is called a cartesian monoidal category.

This leaves open the possibility that there could be a way to make our cartesian category $\mathsf{C}$ into a monoidal category where for each pair $A, B \in \mathsf{C}$ the object $A \otimes B$ equal to the chosen object $A \times B$, but which however is not equivalent, as a monoidal category, to the cartesian monoidal category described above. Is this ever possible?

There are various things we can change: the action of $\otimes$ on morphisms, the associator, the left and right unitors - and prima facie, given the way I've worded the question, even the unit object $I$.

It would be especially fun if we could do this starting from some familiar cartesian category like the category of sets.