  We formulate the full bosonic SL(5) exceptional field theory in a
coordinate-invariant manner. Thereby we interpret the 10-dimensional extended
space as a manifold with $\mathrm{SL}(5)\times\mathbb{R}^+$-structure. We show
that the algebra of generalised diffeomorphisms closes subject to a set of
closure constraints which are reminiscent of the quadratic and linear
constraints of maximal seven-dimensional gauged supergravities, as well as the
section condition. We construct an action for the full bosonic SL(5)
exceptional field theory, even when the
$\mathrm{SL}(5)\times\mathbb{R}^+$-structure is not locally flat.
