Nonlocal curvature without curvature
First think of superfluid helium. The "circulation" vector field v is a "connection" like the EM vector potential A. The vorticity curl v is like magnetic field curlA. Curlv is zero outside of the vortex core in the irrotational superflow. The quantized circulation vortices are multiply connected, i.e. winding numbers or Betti numbers for 1-forms & 2-forms. They are multiply connected because they have "micro-quantum normal fluid cores" where the macro-quantum coherence drops if not to zero to a much smaller value to make a Josephson weak link of sorts. In my macro-quantum theory of gravity for Super Cosmos the "normal fluid" is the exotic vacuum dark energy of negative micro-quantum zero point pressure or the dark matter of positive zero point pressure.
Therefore, we have a Bohm-Aharonov nonlocality in superfluid helium but we do not think it is amazing because we can picture a local circulation vector field of the fluid flow.
Now we have an isomorphism with elasticity theory as shown by Hagen Kleinert. We have a kind of tetrad world crystal elastic distortion field that is essentially related to the non-tensor Levi-Civita connection field (LC), which is like A in EM and like v in superfluid helium. We can, therefore, have non-vanishing "circulation" integrals (line integrals of local g-force) even though there is no interior (covariant) curl of (LC) because the unstable dark energy thin wall is an exotic vacuum "normal fluid" core. In terms of crystal distortion theory, there is a smooth distortion (LC) field outside the source region of a thin unstable dark energy slab inside of which there is Ricci tensor which is a bunch of disclination topological defects that influence at a distance the local (LC) field of vanishing curl.
Now in EM, we say vector potential is not a local observable because it is a inhomogeneous non-tensor under the U(1) local gauge transformations. In GR these are the GCT transition functions between overlapping local coordinate charts in the atlas covering the manifold like analytical continuation of complex functions of complex variable bypassing branch points. (A power series circle of convergence stops at a branch point but you can finagle around the obstruction with a different circle.) Note in my theory the (LC) connection comes from the 4^3 3rd order partial derivatives of the more is different emergent macro-quantum world hologram Goldstone rigid phase of the post-inflationary vacuum coherence field that is the fabric of smooth OLDLRO curved spacetime. So there is no reason at all to say that the (LC) connection is not a local observable even though it is not a GCT tensor.
It's nice, and of course necessary, that the metric as second order partial derivatives of the Goldstone phase and the connection as third order partial derivatives and curvature as 4th order partial derivatives have exactly the correct number of GCT tensor indices required by Einstein's GR!