  We use holography to develop a physical picture of the real-time evolution of
the spinodal instability of a four-dimensional, strongly-coupled gauge theory
with a first-order, thermal phase transition. We numerically solve Einstein's
equations to follow the evolution, in which we identify four generic stages: A
first, linear stage in which the instability grows exponentially; a second,
non-linear stage in which peaks and/or phase domains are formed; a third stage
in which these structures merge; and a fourth stage in which the system finally
relaxes to a static, phase-separated configuration. On the gravity side the
latter is described by a static, stable, inhomogeneous horizon. We conjecture
and provide evidence that all static, non-phase separated configurations in
large enough boxes are dynamically unstable. We show that all four stages are
well described by the constitutive relations of second-order hydrodynamics that
include all second-order gradients that are purely spatial in the local rest
frame. In contrast, a M\"uller-Israel-Stewart-type formulation of hydrodynamics
fails to provide a good description for two reasons. First, it misses some
large, purely-spatial gradient corrections. Second, several second-order
transport coefficients in this formulation, including the relaxation times
$\tau_\pi$ and $\tau_\Pi$, diverge at the points where the speed of sound
vanishes.
