  A long elastic cylinder, radius $a$ and shear-modulus $\mu$, becomes unstable
given sufficient surface tension $\gamma$. We show this instability can be
simply understood by considering the energy, $E(\lambda)$, of such a cylinder
subject to a homogenous longitudinal stretch $\lambda$. Although $E(\lambda)$
has a unique minimum, if surface tension is sufficient
($\Gamma\equiv\gamma/(a\mu)>\sqrt{32}$) it looses convexity in a finite region.
We use a Maxwell construction to show that, if stretched into this region, the
cylinder will phase separate into two segments with different stretches
$\lambda_1$ and $\lambda_2$. Our model thus explains why the instability has
infinite wavelength, and allows us to calculate the instability's sub-critical
hysteresis loop (as a function of imposed stretch), showing that instability
proceeds with constant amplitude and at constant (positive) tension as the
cylinder is stretched between $\lambda_1$ and $\lambda_2$. We use full
nonlinear finite-element calculations to verify these predictions, and to
characterize the interface between the two phases. Near $\Gamma=\sqrt{32}$ the
length of such an interface diverges introducing a new length-scale and
allowing us to construct a 1-D effective theory. This treatment yields an
analytic expression for the interface itself, revealing its characteristic
length grows as $l_{wall}\sim a/\sqrt{\Gamma-\sqrt{32}}$.
