  Bell-inequality violations reveal the presence of quantum correlations
between two particles that have interacted and then separated. Their
generalisation to quantum fields is necessary to study a number of
field-theoretic setups, such as cosmological density fluctuations. In this
work, we show how Bell operators can be constructed for quantum fields in real
space, and for Gaussian states we compute their expectation value in terms of
the field power spectra. We then apply our formalism to a scalar field in
de-Sitter space-time. We find that, in spite of the tremendous production of
entangled particles with opposite wave momenta on large scales, Bell
inequalities are not violated in real space. The reason is that, when
considering measurements of a field at two distinct locations in real space,
one implicitly traces over the configuration of the field at every other
location, leading to a mixed bipartite system. This "effective decoherence"
effect is responsible for the erasure of quantum features, and casts some
doubts on our ability to reveal the quantum origin of cosmological structures.
We finally discuss these results in the light of quantum discord.
