ag.algebraic geometry - Rankin-Selberg convolutions of motivic L-series

If $L(s,M)$ is an irreducible degree 4 motivic L-function, and $L(s,mathrm{sym}^2(M))$ has a pole, then either $M=f_1 otimes f_2$ for a pair of distinct classical modular forms $f_1, f_2$, or $M=mathrm{Asai}(f)$ for $f$ cuspidal on $GL_2(K)$, with $K/mathbb{Q}$ quadratic. You can rule out the Asai case if $L(s,mathrm{sym}^2(M)otimes chi)$ is entire for any nontrivial quadratic character (in particular, entire for $chi$ the character of $K$). If by "practical" you mean "local", then I think the answer is no.