Exceptional zeros of Rankin-Selberg $L$-functions and joint Sato-Tate distributions

Let $χ$ be an idele class character over a number field $F$, and let $π,π'$ be non-dihedral twist-inequivalent cuspidal automorphic representations of $\mathrm{GL}_2(\mathbb{A}_F)$. We prove that if $m,n\geq 0$ are integers, $m+n\geq 1$, $F$ is totally real, $χ$ corresponds with a ray class character, and $π,π'$ correspond with primitive non-CM holomorphic Hilbert cusp forms, then the Rankin--Selberg $L$-function $L(s,\mathrm{Sym}^m(π)\times(\mathrm{Sym}^n(π')\otimesχ))$ has a standard zero-free region with no exceptional Landau--Siegel zero. This is new even for $F=\mathbb{Q}$. As an application, we establish the strongest known unconditional effective rates of convergence in the Sato--Tate distribution for $π$ and the joint Sato--Tate distribution for $π$ and $π'$.