A long elastic cylinder, with radius a and shear-modulus μ, becomes unstable given sufficient surface tension γ. We show this instability can be simply understood by considering the energy, E(λ), of such a cylinder subject to a homogenous longitudinal stretch λ. Although E(λ) has a unique minimum, if surface tension is sufficient [Γ≡γ/(aμ)>√32] it loses 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 λ1 and λ2. Our model thus explains why the instability has infinite wavelength and allows us to calculate the instability's subcritical 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 λ1 and λ2. We use full nonlinear finite-element calculations to verify these predictions and to characterize the interface between the two phases. Near Γ=√32 the length of such an interface diverges, introducing a new length scale and allowing us to construct a one-dimensional effective theory. This treatment yields an analytic expression for the interface itself, revealing that its characteristic length grows as lwall∼a/√Γ−√32.