Are there sentences $\phi$ and $\psi$ in the language of PA and models $\mathcal{M}_\phi$ and $\mathcal{M}_\psi$ of PA such that $\mathcal{M}_\phi\models\phi\wedge(\neg\psi)$ and $\mathcal{M}_\psi\models(\neg\phi)\wedge\psi$, but $\mathbb{N}\models(\neg\phi)\wedge(\neg\psi)$ (where $\mathbb{N}$ is the standard model of PA)?

(I assume that the answer is yes, but I do not know how to construct them or even show non-constructively that they exist.)