We don’t have much trouble yet with relativistic temporal distortions, but Earth’s motion causes us to lose about 0.152 seconds per year relative to the Solar system. Likewise we lose about 8.5 seconds per year relative to the Milky Way. I wonder when we’re going to start to care. Presumably there would be consideration of such issues while dealing with interplanetary spacecraft, timing burns and such.
GPS satellite clocks have to run fast to account for the combined relatavistic effects of moving fast and being significantly farther away from earth's gravity. Without this, they would accumulate around 11km of error per day from losing around 7microseconds per day compared to earthbound clocks.
Earth time <> Sol time <> SagA* time
It is why we are introducing LTC, Coordinated Lunar Time. Apparently the relativistic effects on the Moon are already big enough to make using UTC problematic.