I don't think that's right, docBugs. Say that $V_1$ and $V_2$ are $d$-dimensional and $V_3=\mathbb C^2$. Then an element of $V_1\otimes V_2$ is a linear map $V_1^\star\to V_2$, generically an isomorphism; an element of $V_1\otimes V_2\otimes V_3$ is an ordered pair $(A,B)$ of these; the unordered $d$-tuple of eigenvalues of $B\circ A^{-1}$ is an invariant of the $GL(V_1)\times GL(V_2)$-action; and an element of $GL(V_3)$ will just perform some fractional linear transformation on all of these numbers, so that if $d\ge 4$ then there is a complex invariant here.
I don't think that's right, doc. Say that $V_1$ and $V_2$ are $d$-dimensional and $V_3=\mathbb C^2$. Then an element of $V_1\otimes V_2$ is a linear map $V_1^\star\to V_2$, generically an isomorphism; an element of $V_1\otimes V_2\otimes V_3$ is an ordered pair $(A,B)$ of these; the unordered $d$-tuple of eigenvalues of $B\circ A^{-1}$ is an invariant of the $GL(V_1)\times GL(V_2)$-action; and an element of $GL(V_3)$ will just perform some fractional linear transformation on all of these numbers, so that if $d\ge 4$ then there is a complex invariant here.