  Coarsening of two-phase systems is crucial for the stability of dense
particle packings such as alloys, foams, emulsions or supersaturated solutions.
Mean field theories predict an asymptotic scaling state with a broad particle
size distribution. Aqueous foams are good model systems for investigations of
coarsening-induced structures, because the continuous liquid as well as the
dispersed gas phases are uniform and isotropic. We present coarsening
experiments on wet foams, with liquid fractions up to their unjamming point and
beyond, that are performed under microgravity to avoid gravitational drainage.
As time elapses, a self-similar regime is reached where the normalized bubble
size distribution is invariant. Unexpectedly, the distribution features an
excess of small \textit{roaming} bubbles, mobile within the network of
\textit{jammed} larger bubbles. These roaming bubbles are reminiscent of
rattlers in granular materials (grains not subjected to contact forces). We
identify a critical liquid fraction $\phi^*$, above which the bubble assembly
unjams and the two bubble populations merge into a single narrow distribution
of bubbly liquids. Unexpectedly, $\phi^*$ is larger than the random close
packing fraction of the foam $\phi_{rcp}$. This is because, between
$\phi_{rcp}$ and $\phi^*$, the large bubbles remain connected due to a weak
adhesion between bubbles. We present models that identify the physical
mechanisms explaining our observations. We propose a new comprehensive view of
the coarsening phenomenon in wet foams. Our results should be applicable to
other phase-separating systems and they may also help to control the
elaboration of solid foams with hierarchical structures.
