When people first hear of General Relativity and the idea that spacetime is curved, they naturally imagine a higher-dimensional uncurved space in which our four-dimensional spacetime curvily sits. They may even be shown pictures of a two-dimensional sheet warping into three dimensions. But it's usually then explained to them that that's a misleading way to look at things: reality is just four-dimensional, but has a metric that makes it behave like it's curving in a higher-dimensional space.
What if the natural way of thinking about this is right? What if, say, reality is an 89-dimensional Euclidean space with signature (2,87), but physical objects are constrained to live on a 4-dimensional subset of it? The constraint could be effected, for instance, by a global discontinuous scalar field on the 89-dimensional space that takes two values: 1=allowed and 0=forbidden.
I suppose the main reason not to go for an ontology like this for General Relativity is that it's messier.