  Among the allowed representations of extended supersymmetry in six dimensions
there are exotic chiral multiplets that, instead of a graviton, contain
mixed-symmetry spin-2 tensor fields. Notably, an $\mathcal{N}=(4,0)$ multiplet
has a four index exotic graviton and it was conjectured that an interacting
theory based on this multiplet could arise as a strong coupling limit of M
theory compactified on $T^6$. We present an algebraic study of these multiplets
and their possible embedding into the framework of exceptional field theory,
finding in particular that the six-dimensional momenta do not correspond to a
conventional spacetime section. When compactified on a circle, the
six-dimensional multiplets give rise to the same degrees of freedom as
five-dimensional supergravity theories with the same number of supersymmetries.
However, by considering anomalies (computed using the product multiplets
construction) and the generation of Chern-Simons couplings, we find reason to
doubt that their dynamics will agree with the five-dimensional gravity
theories. We propose an alternative picture, similar to F-theory, in which
particular fixed-volume $T^3$-fibered spacetimes play a central role,
suggesting that only on compactification to three-dimensions will one make
contact with the dynamics of supergravity.
