  We prove the following indistinguishability theorem for $k$-tuples of trees
in the uniform spanning forest of $\mathbb{Z}^d$: Suppose that $\mathscr{A}$ is
a property of a $k$-tuple of components that is stable under finite
modifications of the forest. Then either every $k$-tuple of distinct trees has
property $\mathscr{A}$ almost surely, or no $k$-tuple of distinct trees has
property $\mathscr{A}$ almost surely. This generalizes the indistinguishability
theorem of the author and Nachmias (2016), which applied to individual trees.
Our results apply more generally to any graph that has the Liouville property
and for which every component of the USF is one-ended.
