Can you give us an example of what problem you're trying to solve? I very rarely find myself in need of rounds outside of combat.

@goodguy5 In this answer I discuss using initiative for all members in a split party even when one group is not in combat. Not exactly what you asked, since half the party is still in combat.

I can see using tactical time when part of the table is in a tactical situation and the others may or may not join in, and obviously abilities with a duration in rounds would still happen and last a few seconds, but I'm still not really understanding the context of the question. Like, what kind of things happen in a short rest that might require using rounds to figure out rather than a narrative "a few seconds"? I'd really like to see Illustro's examples.

@DarthPseudonym the idea for this question, and the problem it's looking to solve is the idea of resolving simultaneous effects during longer activities. So for example, a short rest is measured in an hour, a long rest is measures in 8 hours spans of time. Both of these have defined start and end points, and in particular the game specifies a lot of things happen at the end of [insert rest here]. Similarly, what sort of non-combat situations might be suitable for using time tracking in rounds.

@ThomasMarkov no I don't agree, though it's very closely related. That question is looking for rules to use initiative outside of combat (with a particular focus on when to let the party set their own order and when to strictly require initiative rolls). My question, by comparison, is looking slightly higher up, at when should we be tracking time in rounds. Certainly, any time you are using initiative you are definitionally using rounds and turns, but using you can track time in rounds without requiring initiative.

related: When do player characters leave turn-based action (i.e. initiative order) if they are in a hostile area?

@SeriousBri So you need at least one round to resolve multiple "end of long rest" or "end of short rest" effects (since they are all happening at the same time, the end of the rest), as otherwise you can't consistently assign an order to them. The problem isn't the end of the rest per say, but multiple features that all say "at the end of [insert rest period here]" a thing happens needing to have their interactions resolved (as one example of how this happens), but not needing an initiative order to do that resolution

Related: When do player characters leave turn-based action (i.e. initiative order) if they are in a hostile area?, Do you make death saving throws out of combat?, What actions/bonus actions/reactions can a character perform out of combat?

Related: Can a Bonus Action be used outside of a combat encounter?, How does one handle spells out of combat?

I have some notions on how to answer this, but it would be extremely helpful to have a concrete example of your long rest situation: What things happening at the end of a long rest need to have a defined order? Or put another way, what is a situation where the lack of a defined order causes meaningful mechanical ambiguity? (Or short rest, doesn't matter.)

I think explaining how this question came up would significantly improve it. It was certainly from the recent short rest simultaneous effects question, but stating this explicitly would be good

Why do you need any rounds? Time can be measured in many other ways, there are also many things that all end 'at the end of the round' so using rounds in other situations just adds more problems than I can see it solving. I think you might have a question here, but I am not sure it is asked very well, and what I think is your question is a duplicate of the various 'which comes first' questions

@Exempt-Medic Ah, yes, I see. I hate to see a question like this be driven from a weird edge case like that. But in any case, that is substantially different from what I was thinking of.