  We investigate exotic supergravity theories in 6D with maximal (4,0) and
(3,1) supersymmetry, which were conjectured by C. Hull to exist and to describe
strong coupling limits of ${\cal N}=8$ theories in 5D. These theories involve
exotic gauge fields with non-standard Young tableaux representations, subject
to (self-)duality constraints. We give novel actions in a 5+1 split of
coordinates whose field equations reproduce those of the free bosonic (4,0) and
(3,1) theory, respectively, including the (self-)duality relations. Evidence is
presented for a master exceptional field theory formulation with an extended
section constraint that, depending on the solution, produces the (4,0), (3,1)
or the conventional (2,2) theory. We comment on the possible construction of a
fully non-linear master exceptional field theory.
