Is there an algorithmic (or other) way to prove that a (projective) variety is not isomorphic to a toric variety?

I'd be happy with an algebraic answer (for affine or projective varieties), using the fact that toric ideals are binomial prime ideals. There ne could use that the coordinate rings are characterized as those admitting a fine grading by an affine semigroup , i.e. presented by a binomial prime ideal (Prop. 1.11 in Eisenbud/Sturmfels "Binomial ideals").

This question resulted from an Example that I discussed with Mateusz Michalek. The example is: let $V$ be the Zariski closure of the image of the parameterization: $$(p_1,p_2,a_1,a_2,a_3,b_1,b_2,b_3) \to \begin{pmatrix} p_1a_1a_2a_3+p_2b_1b_2b_3 \\ p_1a_1a_2b_3+p_2b_1b_2a_3 \\ p_1a_1b_2a_3+p_2b_1a_2b_3 \\ p_1a_1b_2b_3+p_2b_1a_2a_3 \\ p_1b_1a_2a_3+p_2a_1b_2b_3 \\ p_1b_1a_2b_3+p_2a_1b_2a_3 \\ p_1b_1b_2a_3+p_2a_1a_2b_3 \\ p_1b_1b_2b_3+p_2a_1a_2a_3 \\ \end{pmatrix}$$

Implicitization using Macaulay2 is quick and yields a complete intersection: $$\langle et-ry-qu+wo, wt-qy-ru+eo, we-qr-yu+to \rangle \subset k[q,w,e,r,t,y,u,o]$$

How to prove that $V$ is not toric?