  We prove the rather counterintuitive result that there exist finite
transitive graphs H and integers k such that the Free Uniform Spanning Forest
in the direct product of the k-regular tree and H has infinitely many trees
almost surely.
  This shows that the number of trees in the FUSF is not a quasi-isometry
invariant. Moreover, we give two different Cayley graphs of the same virtually
free group such that the FUSF has infinitely many trees in one, but is
connected in the other, answering a question of Lyons and Peres (2016) in the
negative.
  A version of our argument gives an example of a non-unimodular transitive
graph where WUSF\not=FUSF, but some of the FUSF trees are light with respect to
Haar measure. This disproves a conjecture of Tang (2019).
