Let $T$ be a repetitive tiling (or Delone set) with finite local complexity, and let $\Omega$ be the corresponding tiling space. Then $\Omega$ either contains one Bounded Equivalence (BD) class (if $T$ is BD to a lattice) or uncountably many (if it isn't). This is joint work with Alexey Garber and Dirk Fretloeh. This result was also proved independently (in somewhat greater generality!) by Yotam Smilansky and Yaar Solomon.