= Solution
The recursive theory $T=PA^-+\varphi$ is consistent because $\mathbb N\models T$. By the Gödel-Rosser theorem it has an undecidable sentence $\rho$, so both $T+\rho$ and $T+\neg\rho$ are consistent. Exactly one of $\rho,\neg\rho$ is false in $\mathbb N$; add that one to $T$. The first-order completeness theorem gives a model, and the <Downward Lowenheim-Skolem theorem> gives a countable model $M$. Then $M\models PA^-+\varphi$, but $M$ disagrees with $\mathbb N$ on the chosen sentence and is therefore not elementarily equivalent to it.
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