We construct a pseudo-Lagrangian that is invariant under rigid
E11and transforms as a density under E11generalised diffeomorphisms. The gauge-invariance requires the use of a section condition studied in previous work on E11exceptional field theory and the inclusion of constrained fields that transform in an indecomposable E11-representation together with the E11coset fields. We show that, in combination with gauge-invariant and E11-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 condi- tion. For another choice, we reobtain the E8exceptional 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 E10sigma model.