  Binary black holes with spins that are aligned with the orbital angular
momentum do not precess. However, post-Newtonian calculations predict that
"up-down" binaries, in which the spin of the heavier (lighter) black hole is
aligned (antialigned) with the orbital angular momentum, are unstable when the
spins are slightly perturbed from perfect alignment. This instability provides
a possible mechanism for the formation of precessing binaries in environments
where sources are preferentially formed with (anti) aligned spins. In this
paper, we present the first full numerical relativity simulations capturing
this instability. These simulations span $\sim 100$ orbits and $\sim 3$-$5$
precession cycles before merger, making them some of the longest numerical
relativity simulations to date. Initialized with a small perturbation of
$1^{\circ}$-$10^{\circ}$, the instability causes a dramatic growth of the spin
misalignments, which can reach $\sim 90^{\circ}$ near merger. We show that this
leaves a strong imprint on the subdominant modes of the gravitational wave
signal, which can potentially be used to distinguish up-down binaries from
other sources. Finally, we show that post-Newtonian and effective-one-body
approximants are able to reproduce the unstable dynamics of up-down binaries
extracted from numerical relativity.
