  Quantum gravitational effects can become important at low energy if the
wavefunction of the metric field fails to be peaked around a classical
configuration. We try to understand such deviations from classicality within
canonical quantum gravity by introducing a "fluid of observers" in the low
energy theory and defining a distance operator "at equal time" among them. We
find that, even in the presence of relevant fluctuations in the metric field, a
locally flat limit is recovered in the neighbourhood of each observer.
Deviations from classicality have no particular consequence, locally. However,
at larger separations the expectation value of the distance operator behaves
differently than a standard Riemannian distance. In particular, it is
non-additive and thus cannot be obtained by the integral of a differential line
element. This emerging "beyond Riemannian" geometry is a metric space similar
to embedded Riemannian manifolds equipped with chord distances that cut through
the ambient space. We study deviations from flat space by looking at triangles
in the limit where their sizes go to zero. Beyond-Riemannian deviations with
respect to flat space are of the same order as standard Riemannian ones, but
qualitatively different. Possible connections with holography and with the
black hole information paradox are briefly discussed.
