  We give a two variable Rankin-Selberg integral inspired by consideration of
Garrett's pullback formula. For a globally generic cusp form on
$\mathrm{GL}_2\times \mathrm{GSp}_4$, the integral represents the product of
the $\mathrm{Std}\times \mathrm{Spin}$ and $\mathbf{1} \times \mathrm{Std}$
$L$-functions. We prove a result concerning an Archimedean principal series
representation in order to verify a case of Jiang's first-term identity
relating certain non-Siegel Eisenstein series on symplectic groups. Using it,
we obtain a new proof of a known result concerning possible poles of these
$L$-functions.
