It's called a black hole merger, or coalescence. Here a simulation video.
Even the formation of the event horizons of the two initial black holes takes "super long" in Earth's time. That's similar with the merger. On the other hand we are very close to the completed merger within a short "Earth's" time seen from a distance, as soon as the merger starts. General relativity as well as quantum theory are incomplete with what will happen very close to a presumed singularity or at the presumed event horizon; this will remain disputed until a satisfying theory of quantum gravity is found.
Mergers of black holes are likely to occur, e.g. when two galaxies collide, the momentum of the central supermassive black holes (SMBH) is slowed down by consumption of gas, dust and stars, until the SMBHs merge to the central SMBH of the merged galaxy.
Here a galaxy merger simulation.
Here a simulation of the coalescence of two black holes within a collapsing star.
More on black hole binaries on Wikipedia.