Calling half time and changing ends is a nice idea, but utterly impractical. Finding/building an alternative neutrino source and pointing it at the detector is (I think) the best approach to reproducibility, but it still leaves open the possibility of systematic errors in the detection mechanism causing the same issues. A different distance should give a different delay though, which would be one test.
Another might be to try and use photons as a reference/synchronisation mechanism, but that's hindered by the fact that everything is in underground caverns.
Yes it would probably cost millions for such a thing to be done if it's even possible.
Figuring out the appropriate clock corrections very, very carefully seems like a better approach...
I wonder how accurate the GPS synchronization is too. I'm wondering if it takes into account different atmospheric conditions and the index of refraction of radio waves in air.
This probably wouldn't work, time dilation under gravity makes my head hurt!
how about just inside the fiber. How good is the ping between CERN and Gran Sasso ? :)