  A "tensor space" is a vector space equipped with a finite collection of
multi-linear forms. In previous work, we showed that (for each signature) there
exists a universal homogeneous tensor space, which is unique up to isomorphism.
Here we generalize that result: we show that each Zariski class of tensor
spaces contains a weakly homogeneous space, which is unique up to isomorphism;
here, we say that two tensor spaces are "Zariski equivalent" if they satisfy
the same polynomial identities. Our work relies on the theory of
$\mathbf{GL}$-varieties developed by Bik, Draisma, Eggermont, and Snowden.
