  It is broadly known that any parallelepiped tiles space by translating copies
of itself along its edges. In earlier work relating to higher-dimensional
sandpile groups, the second author discovered a novel construction which
fragments the parallelpiped into a collection of smaller tiles. These tiles
fill space with the same symmetry as the larger parallelepiped. Their volumes
are equal to the components of the multi-row Laplace determinant expansion, so
this construction only works when all these signs are non-negative (or
non-positive).
  In this work, we extend the construction to work for all parallelepipeds,
without requiring the non-negative condition. This naturally gives tiles with
negative volume, which we understand to mean canceling out tiles with positive
volume. In fact, with this cancellation, we prove that every point in space is
contained in exactly one more tile with positive volume than tile with negative
volume. This is a natural definition for a signed tiling.
  Our main technique is to show that the net number of signed tiles doesn't
change as a point moves through space. This is a relatively indirect proof
method, and the underlying structure of these tilings remains mysterious.
