  We give a two-variable Rankin--Selberg integral for generic cusp forms on
$\mathrm{PGL}_4$ and $\mathrm{PGU}_{2,2}$ which represents a product of
exterior square $L$-functions. As a residue of our integral, we obtain an
integral representation on $\mathrm{PGU}_{2,2}$ of the degree 5 $L$-function of
$\mathrm{GSp}_4$ twisted by the quadratic character of $E/F$ of cuspidal
automorphic representations which contribute to the theta correspondence for
the pair $(\mathrm{PGSp}_4,\mathrm{PGU}_{2,2})$.
