= Solution
Consistency of $PA^-$ makes $A$ and $B$ disjoint. Suppose a recursive set $C$ separated them, and let $R(x)$ represent its total characteristic function in $PA^-$. By the <Diagonal lemma>, choose a sentence $\sigma$ satisfying
$$
PA^-\vdash\sigma\leftrightarrow\neg R(\ulcorner\sigma\urcorner).
$$
Put $n=\ulcorner\sigma\urcorner$. If $n\in C$, representability gives $PA^-\vdash R(\bar n)$ and hence $PA^-\vdash\neg\sigma$, so $n\in B$, contradicting $B\cap C=\varnothing$. If $n\notin C$, representability gives $PA^-\vdash\neg R(\bar n)$ and hence $PA^-\vdash\sigma$, so $n\in A\subseteq C$, again a contradiction. Therefore $A$ and $B$ are recursively inseparable.
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