Are there sentences $\phi$ and $\psi$ in the language of PA and models $\mathcal{M}_\phi$ and $\mathcal{M}_\psi$ of PA such that $\mathcal{M}_\phi\models\phi$ and $\mathcal{M}_\psi\models\psi$, but $\text{PA}\vdash\neg(\phi\wedge\psi)$ and $\mathbb{N}\models(\neg\phi)\wedge(\neg\psi)$ (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. This answer seems to suggest that $\phi :\leftrightarrow \neg\text{Con}(\text{PA})\wedge\text{Con}(\text{PA}+\neg\text{Con}(\text{PA}))$ and $\psi :\leftrightarrow \neg\text{Con}(\text{PA})\wedge\neg\text{Con}(\text{PA}+\neg\text{Con}(\text{PA}))$ should work, but it's not obvious to me from the form of the second incompleteness theorem that I'm familiar with that $\phi$ is consistent: $\text{PA}+\neg\text{Con}(\text{PA})$ cannot prove its own consistency, but is it clear that it can't prove its own inconsistency either?
