Angular momentum orientation of the orbits of the two stars with respect to the Solar System Happen to see this in Wiki...
The astronomer Karl Schwarzschild observed the motion of pairs of stars orbiting each other. He found that the two orbits of the stars of such a system lie in a plane, and the perihelion of the orbits of the two stars remains pointing in the same direction with respect to the solar system. Schwarzschild pointed out that that was invariably seen: the direction of the angular momentum of all observed double star systems remains fixed with respect to the direction of the angular momentum of the Solar System. These observations allowed him to conclude that inertial frames inside the galaxy do not rotate with respect to one another, and that the space of the Milky Way is approximately Galilean or Minkowskian.[59]
My question is, are there any observations of binary star systems that violate the angular momentum orientation, so far?
 A: There are lots of observations that show that the line of apsides- the line joining the periapsis points of two stars - changes with time.
The phenomenon is known as apsidal precession.
Apsidal precession can of course be caused by $>2$ bodies in a system, but even for the 2-body systems discussed in the question, there are two mechanisms that lead to apsidal precession - it is predicted by General Relativity and it happens if the bodies are non-spherical and have a quadrupole moment.
In binary star systems, apsidal precession is well established in objects like the Hulse-Taylor binary, where the line of apsides rotates by 4.2 degrees per year.
None of this contradicts conservation of angular momentum - the precession of the apsides is in the orbital plane.
A: Before I get to the main question: the statements in the wikipedia article about what Schwarzschild did are incorrect (but not in a way that touches on the question submitted here).
The wikipedia article has a reference to the source: a publication titled: In the Shadow of the Relativity Revolution
The third section is by Mattias Schemmel and is titled:
An Astronomical Road to General Relativity:
The Continuity between Classical and Relativistic Cosmology
in the Work of Karl Schwarzschild
Quote from page 43:
Schwarzschild had to defend five theses, probably formulated by his teacher von Seeliger, in order to obtain his post-doctoral degree in 1899. One of these theses read: "The existence of centrifugal forces is comprehensible only under the assumption of a medium pervading all of space."
Quote from page 44:
From the observation that the perihelia of double stars are at rest with respect to the directions that seem fixed inside the solar system, Schwarzschild concludes that the directions distinguished in their region of space have to be the same as in the solar system.

Clearly Schwarzschild was drawing upon an existing body of observational data, not on new observations of his own.


With the above out of the way:
No.
If at some point in time such a violation of angular momentum orientation would be observed then it would count as a major anomaly. It would count as an inexplicable observation.
If at some point in time such a thing would have been observed every physicist would soon know about it.
