I'm in trouble with the definition of reference frame in the context of General Relativity intended as coordinate chart (i.e. no frame field).
Various sources define it as a smooth map from an open region of spacetime into an open region of $\mathbb R^4$ endowed with the standard euclidean topology.

Now my point is: in any specific circumstance, e.g. Schwarzschild spacetime, in order to assign coordinates to events a sort of whatsoever  "rule" is actually needed. Following for instance the point made by Landau & Lifshitz in The classic Theory of Fields - $82

> This result essentially changes the very concept of a system of
> reference in the general theory of relativity, as compared to its
> meaning in the special theory. In the latter we meant by a reference
> system a set of bodies at rest relative to one another in unchanging
> relative positions. Such systems of bodies do not exist in the
> presence of a variable gravitational field, and for the exact
> determination of the position of a particle in space we must, strictly
> speaking, have an infinite number of bodies which fill all the space
> like some sort of "medium". **Such a system of bodies with arbitrarily
> running clocks fixed on them constitutes a reference system in the
> general theory of relativity**.

So, a coordinate chart isn't just a mathematical abstract tools, it (at least) tacitly includes/implies the rules used to map events to points in the map's image (an open set of $\mathbb R^4$).