For finance, it matters what order events occur in. I've never messed with control systems, but I suppose similar issues arise.

Control systems, or any form of embedded safety critical system, use monotonic clocks where calendar time is a complete non-issue.

Ordering of events, on what scale are we talking here? If it’s just within a transactional database there are multitude of ways to do it. Even distributed dbs have such features without relying on perfect time. If you are looking at a spanner style db you need a lot more guarantees than “I just used the time my cloud provider assigned to my vm”, plus being in sync only matters within your own cluster?

Ordering can be solved by a mutex.

I was thinking of more distributed control systems, particularly where testing for edge cases might be difficult and rare, and rigorous methods (did Lamport solve distributed mutex?) are probably off the radar in terms of culture.

Any forum is a distributed system of a server and clients that agree with each other on order of user posts.