Every instrument we use to "measure" time involves movement: pendulum, sandglass, rotation of Earth, translation of Earth around the Sun, vibration of atoms, etc.
So some people (like Rovelli) think one should try to formulate relativity only in terms relations of events. This might be relevant in a more fundamental theory of spacetime but on a classical level it’s irrelevant. You can measure the geometry of spacetime in our solar system w/o disturbing it. The gravitational field of e.g. some satellite is negligible to the gravitational field of planets and the sun.
Thus, the "reality" is that no two clocks are in sync. As Rovelli points out in "Reality Is Not What It Seems", with the precise clocks we have today, it's possible to measure this effect for a difference in altitude of a few centimetres.