First, I apologize for the long preface to my question. The actual question is in bold below.
In Kardar's Statistical Physics of Fields, on page 10, the author writes, "the free energy of the system is an analytical function in the $(P,T)$ plane, except for a branch cut along the phase boundary." While the above comment is regarding the liquid-gas transition, the same idea is described in the para/ferromagnetic phase transition. In each case, a first order phase transition corresponds to passing through a branch cut, while a second order phase transition occurs exactly at the branch point.
In some ways, describing the phase transition as passing through a branch cut in parameter space makes a lot of sense: namely, you expect your equilibrium observables to vary discontinuously when approaching the branch cut from each side. On the other hand, in complex analysis, branch cuts are usually used when discussing functions which are multivalued (ie, must be defined on multiple Riemann sheets). In this sense, describing the free energy (or other observables) as having branch cuts does not make sense: equilibrium observables are necessarily single-valued, and therefore one could not "push up" the branch cut and explore an additional Riemann sheet.
However, this line of thought does raise an interesting question. Suppose I start in some equilibrium state, and adiabatically vary my parameters so as to explore different equilibrium configurations -- for concreteness, let's say I'm working with an Ising model, so that I can vary magnetic field $h$ and temperature $T$. If I were to start with $h > 0$ and $T < T_c$, and I adiabatically lowered $h$ to pass through zero, Is there some possibility to explore some sort of metastable state with free energy, magnetization, etc, an analytic function of $h$ as $h$ passes through zero? Heuristically, I'm imagining that if we start in the completely ordered spin-up state, for a small field in the "down" direction, each individual spin would rather continue to align upwards with its neighbors instead of flipping downwards towards the field. Of course, much is missing to make this argument rigorous.