  If ultralight bosonic fields exist in Nature as dark matter, superradiance
spins down rotating black holes (BHs), dynamically endowing them with
equilibrium bosonic clouds, here dubbed synchronised gravitational atoms
(SGAs). The self-gravity of these same fields, on the other hand, can lump them
into (scalar or vector) horizonless solitons known as bosonic stars (BSs). We
show that the dynamics of BSs yields a new channel forming SGAs. We study BS
binaries that merge to form spinning BHs. After horizon formation, the BH spins
up by accreting the bosonic field, but a remnant lingers around the horizon. If
just enough angular momentum is present, the BH spin up stalls precisely as the
remnant becomes a SGA. Different initial data lead to SGAs with different
quantum numbers. Thus, SGAs may form both from superradiance-driven BH spin
down and accretion-driven BH spin up. The latter process, moreover, can result
in heavier SGAs than those obtained from the former: in one example herein,
$\sim 18\%$ of the final system's energy and $\sim 50\%$ of its angular
momentum remain in the SGA. We suggest that even higher values may occur in
systems wherein both accretion and superradiance contribute to the SGA
formation.
