  We construct a pseudo-Lagrangian that is invariant under rigid $E_{11}$ and
transforms as a density under $E_{11}$ generalised diffeomorphisms. The
gauge-invariance requires the use of a section condition studied in previous
work on $E_{11}$ exceptional field theory and the inclusion of constrained
fields that transform in an indecomposable $E_{11}$-representation together
with the $E_{11}$ coset fields. We show that, in combination with
gauge-invariant and $E_{11}$-invariant duality equations, this
pseudo-Lagrangian reduces to the bosonic sector of non-linear
eleven-dimensional supergravity for one choice of solution to the section
condition. For another choice, we reobtain the $E_8$ exceptional field theory
and conjecture that our pseudo-Lagrangian and duality equations produce all
exceptional field theories with maximal supersymmetry in any dimension. We also
describe how the theory entails non-linear equations for higher dual fields,
including the dual graviton in eleven dimensions. Furthermore, we speculate on
the relation to the $E_{10}$ sigma model.
