I am trying to understand the relationship between gauge theories (for example General Relativity) and the well-posedness of the underlying theory. In General Relativity, it is known that you must be careful in choosing your gauge (i.e. your coordinates) in order for the resulting PDE system to be well-posed. This is a current problem in many alternative theories of gravity.
The reason I am asking this here is that I would like to gain some mathematical intuition about this. The above seems to say that contrary to the popular saying, Nature does care at least a little bit about our coordinate system. Is this true only for gauge theories like GR or more generally non-linear PDEs? Could I for example find some weird coordinate system to make the advection equation "manifestly" ill-posed? If this sort of problem occurs only for gauge theories, is there an intuitive way to understand this?
In principle, you can write down any PDE you wish, as long as you keep writing equations in a real space/set such that it makes sense. If the "well-posed-ness" includes solvability in the sense of the existence and uniqueness of solutions to a PDE, this becomes a lot more restrictive. I think working this out is one of the hardest tasks when studying a certain PDE, and this is merely understood to a certain extend for very special classes of PDE's.
One example coming to my mind is the Klein-Gordon equation (which is more or less the easiest useful field equation) on an arbitrary spacetime. If you want to impose solvability, you end up with a very special class of space-times, so-called globally hyperbolic space-times, where you can actually show that the equation always admits a unique solution for a prescribed set of initial values, say on a time slice submanifold. However, this property is basically equivalent to the space-time being of the form $\mathbb{R}\times M$ with some manifold $M$ and a metric $-\text{d}t^2+ g_M(t)$ such that $g_M(t)$ turns $M$ into a Riemannian manifold. This is pretty restrictive and a very symmetric setting.
And a remark on "the popular saying, Nature does care at least a little bit about our coordinate system": I have not heard of it, but in my opinion this is just false by the exact same reason why we call a gauge a "gauge". You want to never have a physical outcome to depend on a special gauge you needed to predict that outcome in your formulas. If you are able to show existence and uniqueness formally in a certain coordinate system, then what should prevent your solution to exist or to be unique in another coordinate system, at least on the set which was covered by the previous coordinate system?