  Decoherence is ubiquitous in quantum physics, from the conceptual foundations
to quantum information processing or quantum technologies, where it is a threat
that must be countered. While decoherence has been extensively studied for
simple, well-isolated systems such as single atoms or ions, much less is known
for many-body systems where inter-particle correlations and interactions can
drastically alter the dissipative dynamics. Here we report an experimental
study of how spontaneous emission destroys the spatial coherence of a gas of
strongly interacting bosons in an optical lattice. Instead of the standard
momentum diffusion expected for independent atoms, we observe an anomalous
sub-diffusive expansion, associated with a universal slowing down $\propto
1/t^{1/2}$ of the decoherence dynamics. This algebraic decay reflects the
emergence of slowly-relaxing many-body states, akin to sub-radiant states of
many excited emitters. These results, supported by theoretical predictions,
provide an important benchmark in the understanding of open many-body systems.
