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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.