  We prove a reciprocity formula for the average of the product of
Rankin--Selberg $L$-functions
$L(1/2,\Pi\times\widetilde{\sigma})L(1/2,\sigma\times\widetilde{\pi})$ as
$\sigma$ varies over automorphic representations of $\mathrm{PGL}(n)$ over a
number field $F$, where $\Pi$ and $\pi$ are cuspidal automorphic
representations of $\mathrm{PGL}(n+1)$ and $\mathrm{PGL}(n-1)$ over $F$,
respectively. If $F$ is totally real, and $\Pi$ and $\pi$ are tempered
everywhere, we deduce simultaneous non-vanishing of these $L$-values for
certain sequences of $\sigma$ with conductor tending to infinity in the level
aspect and bearing certain local conditions.
