  Rotating black holes can produce superradiant clouds of ultralight bosons.
When the black hole is part of a binary system, its cloud can undergo
resonances and ionization. These processes leave a distinct signature on the
gravitational waveform that depends on the cloud's properties. To determine the
state of the cloud by the time the system enters the band of future millihertz
detectors, we study the chronological sequence of resonances encountered during
the inspiral. For the first time, we consistently take into account the
nonlinearities induced by the orbital backreaction and we allow the orbit to
have generic eccentricity and inclination. We find that the resonance
phenomenology exhibits striking new features. Resonances can "start" or "break"
above critical thresholds of the parameters, which we compute analytically, and
induce dramatic changes in eccentricity and inclination. Applying these results
to realistic systems, we find two possible outcomes. If the binary and the
cloud are sufficiently close to counter-rotating, the cloud survives in its
original state until the system enters in band; otherwise, the cloud is
destroyed during a resonance at large separations, but leaves an imprint on the
eccentricity and inclination. In both scenarios, we characterize the
observational signatures, with particular focus on future gravitational wave
detectors.
