  We investigate the loss of spatial coherence of one-dimensional bosonic gases
in optical lattices illuminated by a near-resonant excitation laser. Because
the atoms recoil in a random direction after each spontaneous emission, the
atomic momentum distribution progressively broadens. Equivalently, the spatial
correlation function (the Fourier-conjugate quantity of the momentum
distribution) progressively narrows down as more photons are scattered. Here we
measure the correlation function of the matter field for fixed distances
corresponding to nearest-neighbor (n-n) and next-nearest-neighbor (n-n-n) sites
of the optical lattice as a function of time, hereafter called n-n and n-n-n
correlators. For strongly interacting lattice gases, we find that the n-n
correlator $C_1$ decays as a power-law at long times, $C_1\propto
1/t^{\alpha}$, in stark contrast with the exponential decay expected for
independent particles. The power-law decay reflects a non-trivial dissipative
many-body dynamics, where interactions change drastically the interplay between
fluorescence destroying spatial coherence, and coherent tunnelling between
neighboring sites restoring spatial coherence at short distances. The observed
decay exponent $\alpha \approx 0.54(6) $ is in good agreement with the
prediction $\alpha=1/2$ from a dissipative Bose-Hubbard model accounting for
the fluorescence-induced decoherence. Furthermore, we find that the n-n
correlator $C_1$ controls the n-n-n correlator $C_2$ through the relation $C_2
\approx C_1^2$, also in accordance with the dissipative Bose-Hubbard model.
