What is the topology of a phase diagram? This question is cross-posted from phys

Looking at various two-variable phase diagrams I was struck by that on every one I have seen so far all the phases formed simple connected regions; see, for example the phase diagrams of $\ce{H2O}$ or of $\ce{Fe-Fe_3C}$ in 1. Every phase is a connected region, no two disjoint pieces correspond to the same phase. Is this really true in general and if so why?

Going further, let us consider a three-variable phase diagram, say $p, T, y$, where $y$ maybe electric or magnetic field intensity, or some other controllable extensive/intensive variable. Is it possible that now a 3d region occupied by some phase is a kind of tube that makes a half turn so that if we fix, say, $y$ then the resulting 2d section would have two disjoint $p,T$ regions of the same phase? Can such phase "tube" exist or by some thermodynamic-geometric rule is it excluded? 

A possibly more difficult question if a phase can form an annular  region, ie., homeomorphic to an annulus while surrounding another phase homeomorphic to a disk? 

In short, what is the topology of the phase diagram and its sections?
Particularly since Fe was mentioned, I'll focus there. For pure iron, it is well known that $\alpha$-Fe, the stable phase at room temperature, is bcc. This is stable up to 1184K where it transitions to fcc $\gamma$-Fe. In turn, this transforms back to bcc $\delta$-Fe at 1667K before melting at 1810K. Yet, both $\alpha$ and $\delta$ have a bcc crystal structure. 

If one looks at the Fe-Zn phase diagram:

one sees that the region of bcc phase, enclosed in dark blue, fully encompasses the green sliver of fcc $\gamma$. Using the Zn concentration as a free parameter, one can go from $\alpha$ to $\delta$ without changing phase. 

In contrast, in the Fe-Pd phase diagram:

one sees a different picture. Here, Fe and Pd show complete miscibility in the fcc phase across the entire diagram. This means you can't go from $\alpha$ to $\delta$ bcc phase without going through the fcc phase. Yet, one can see that, by trading off Zn for Pd, one could manage to get around the fcc phase by some path in composition space.