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?

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