The 3 important anomalies in Physics in the last 3 decades can be addressed as a null effect if one is to use either Theory Of Relativity alone or a conjunction of Theory of Relativity with Quantum Mechanics when the latter is imminent; eg in case of OPERA we deal with Quantum-Particles such as photons and neutrinos. This is described in this short article written 2 years ago when the OPERA anomaly was doing rounds in every corner of Physics-World.
The very nature of earth-based gravitational effects is, at-least a fallout in the order of mm/second and application of Quantum Mechanics in case of OPERA apparatus brings out this inherent truth clearly.
One would see at minimum a (+ or -) mm/second order excess beyond speed-of-light = 1, no matter what. On the other hand flyby “anomaly” (which is rather a puzzle) is consistent with basic Relativity without need of Quantum Mechanics. The OPERA “anomaly” (which is basically a puzzle created out of carelessness) is based on an observation or more correctly a “misinterpretation” of 7.5 km/second but using Quantum Mechanics one sees that one should only observe ~5 mm/second given we have a fixed uncertainty on time-measurement to 1 nano-second level.
The mm/second is the order of anomaly observed in Flyby and even Pioneer satellites. With the correct interpretation one sees with or without application of Quantum Mechanics, such anomalies are consistent with theory of special and general Relativity. Relativity is a mm level accurate theory for objects of the size of earth, the range of gravity starts at mm level and goes to infinite separation.
A fact often not told because it has perhaps never been brought forth to anyone’s notice (by a dint of magic that Physics is not?). This we can mention as an issue in the unification of gravity with Quantum Mechanics, because the higher range of infinite separation is fine but the lower limit range makes anything below mm or mm/second a difficult paradigm to deal with when Quantum Mechanics deals with ranges that are far below mm.
If this is only an observational or experimental issue one has to look beyond to great distances in cosmology. That has been very very difficult and you will find such mentions clearly, in the literature.