(This is too long for a comment.) This follows from the fact that $\text{SL}_n\mathbb{Z}$ is Zariski dense in $\text{SL}_n\mathbb{R}$. Let $\phi:V\to W$ be a linear isomorphism such that for every $g\in \text{SL}_n\mathbb{Z}$, the "conjugate" $\phi^g = g^{-1}\cdot\phi(g\cdot -)$ equals $\phi$. The subset of $\text{SL}_n\mathbb{R}$ of all $g$ such that $\phi^g$ equals $\phi$ is the simultaneous zero locus of finitely many polynomial functions of the matrix entries of $g$: just look at the matrix entries of $\phi^g - \phi$ as functions of $g$. Thus, since these polynomials vanish on the Zariski dense subset $\text{SL}_n\mathbb{Z}$, they vanish on all of $\text{SL}_n\mathbb{R}$.