Are there solutions of general relativity for any realistic physical problem?

https://en.wikipedia.org/wiki/Exact_solutions_in_general_relativity

The dictatorial strictness of the Birkhoff equation bans any matter field lagrangian L_M which would create or lose energy. The solution outside the spherically symmetric system is a static Schwarzschild metric with a constant gravitating mass M, regardless of what spherically symmetric physical process happens at the origin.

The strictness hints that there actually might only exist solutions of general relativity in artificial, highly symmetric setups.

For example, can we glue together two Schwarzschild solutions and obtain a solution for the two body problem? We certainly can solve such an initial condition problem in newtonian mechanics. If we would have a rubber sheet model of general relativity, the existence of a solution is obvious, though the existence may be hard to prove.

Let us consider a geometric problem with a smooth 2D manifold (surface) in the euclidean 3D space.

Suppose that the Ricci scalar curvature in the manifold is zero everywhere but at two circular areas, where it is a constant R_0.

If we would have just one circular area with R_0, then the solution is a cone (?) or a tube where the circular area functions as the cap.

Can we somehow glue together two cones, so that the Ricci scalar is zero at the gluing point? It looks hard, except in a few symmetric cases.
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Above is a solution: a pipe with two half-sphere caps. Can we add still a third circular area with R_0? That looks hard.

Einstein's field equations imply that the Ricci scalar curvature R is zero everywhere where the stress-energy tensor T is zero. That greatly restricts the geometry of the solution.

In a rubber model, there would probably not be such a strict restriction on R.

Two static masses held at a constant separation

Has anyone calculated the general relativity solution for two masses which are kept static and at a constant distance by electric repulsion? Or alternatively, are kept at the distance by a rigid rod?

Can we glue together two individual solutions, such that the Ricci scalar curvature is exactly what is required by the energy of the static electric field, or by the mass and the pressure in the rod?

The newtonian solution gives a hint. Suppose that we shoot an expanding ball of dust at a velocity v from the middle point between the masses. Does the dust ball volume keep growing at the same speed as it would in a Minkowski space, or is it "focused" or "defocused" as it grows? The newtonian approximation is linear. The behavior of the dust ball is obtained by summing the effect of the fields of the two masses. Since the Ricci scalar is zero for each field independently, it is zero for the linear sum.

If the distance of the two masses is of the order of the Schwarzschild radius r_s of each mass, then the problem is harder. Gravity is not linear in this case. Could it be that the dust ball gets focused or defocused in such an extreme gravitational field?

Is Birkhoff's theorem compatible with a wormhole?

Suppose that we are able to feed more energy to the spherically symmetric system through a symmetric wormhole which opens there at the origin. Then we could increase the gravitating mass M of the system. Birkhoff's theorem would be broken.

Are there solutions for the whole universe?

Suppose that the universe is approximately a finite balloon and is expanding or contracting. Birkhoff's theorem is a symptom that general relativity is not flexible in finding solutions. Are there any solutions for the case where the mass distribution is not uniform?

https://www.ncbi.nlm.nih.gov/pmc/articles/PMC5255525/

A survey paper by Alan D. Rendall lists a large amount of research into the existence of solutions of general relativity. At a first glance, it seems to be an open problem if a global solution for the universe exists in any realistic case.

Are there realistic solutions for an asymptotically Minkowski space?

There seems to be a lack of non-existence theorems, too.

The appearance of singularities in a collapse might be interpreted as a non-existence theorem, if we ban solutions with singularities. Apparently, it is an open problem if singularities form in a realistic collapse.