Define Implicit Region using Interpolating Function

Suppose I have a set of points ${(x_1,x_2) : g(x_1,x_2) > f(x_1,x_2)}$, how would I turn this into a mesh that acts as a region for PDE solvers, or even to use as a plotting region?

I have tried using Implicit Region, but my function $f$ is an interpolating function and so is defined over a Disk region, which is not a square domain.

Hence if I run

omega = ImplicitRegion(g(x1,x2) > f(x1,x2), {{x1, -1, 1}, {x2, -1, 1}})

and try to plot my region

Region(omega)

It doesn’t give me anything, and trying to use this omega for use as a plotting domain / PDE domain doesn’t work either. However, I can plot my region by running

set = f(x1,x2) > g(x1,x2)
RegionPlot(set, {x1, -1, 1}, {x2, -1, 1})

I suppose this means RegionPlot is more robust in the sense that it doesn’t care if my function $f$ is indeterminate at certain points.

I have also tried to change the values of $x_1,x_2$ in the implicit region definition to where my function $f$ is defined i.e. commands like

omega = ImplicitRegion(g(x1,x2) > f(x1,x2), {{x1,x2} in Disk(0,0,r)})

dont seem to work for ImplicitRegion.

Hope my question is clear, happy to clarify further.