Suppose that $\mathcal{C}$ is a model category (with a fixed model structure). Are there any known examples where $\mathcal{C}$ is a (symmetric) monoidal model category in two different ways? I.e., can there exist tensor products $\otimes_1, \otimes_2\colon \mathcal{C}\times \mathcal{C}\to \mathcal{C}$ such that both $(\mathcal{C}, \otimes_1, A)$ and $(\mathcal{C}, \otimes_2, B)$ are symmetric monoidal model categories. Can this phenomenon even occur?

The canonical place to look for an answer to this would be Balchin's book A handbook of Model Categories (specifically in the Die Kunstkammer) but I couldn't see anything relevant.