The movie does a variation on this exact plot. Very well, even.
("Great minds think alike.")
For anything legal, scientific, etc., dealing with time or speed (tracking experiments, wage and hour laws, etc.) you need to be able to track time during the clock pause, which will lead to adoption (by social convention, if nothing else) of a standard method of recording times within the pause, which leads to clocks that display in that format times during the pause. “All clocks stop for 40 minutes” is not a stable equilibrium.
Source: 2 years of heavy Windows 10/11 usage with that setting.
So TAI is not the end all for what I expect to be a very complicated technical problem.
We already have that sort of sync happening today: when your computer is off, it runs a quartz watch. When you turn it on, it syncs via ntp.
Some special boffins on Mars would run atomic clocks, and worry about synchronisation with earth.
For the rest of the people running quartz clocks, they wouldn't even notice.
I think at some point, you have to redefine utc to be based on an estimated atomic clock at the center of the sun, because it's easier for mars and earth to estimate that, than for earth to go on its merry way and mars to estimate the time on earth. Relativism is fun?
The biggest effect is the Sun's gravity, followed by orbital velocity. The relative masses of the planets is negligible by comparison, though in online discussions that's almost all anybody ever seems to talk about.
I can't find confirmation through other sources, and can't credibly validate it myself, but the best estimate I found for time dilation between two clocks on Earth and Mars is roughly 5.6 parts per billion, or ~177 milliseconds per year. See https://space.stackexchange.com/questions/33590/what-is-the-...
And when the clocks go back in autumn you get the 40 minute period twice.