Clocks would display both a "World Time" and a "Zone Time".
The World Time would probably simply be UTC.
For coordinating across significant longitude differences World Time would be used.
The Zone Time applies to a region similar to a current time zone, and is an idealized solar time, adjusted so that sunrise each day is around 6:00 AM LZT (Local Zone Time).
Schools, stores, most offices, mass transit routes that are mostly in the zone, theaters, meals, and similar things would base their schedules on LZT. So office hours, for example, would typically be 9:00 AM to 5:00 PM LZT, year around.
This would let most people work on a schedule that syncs well to the Sun, which should have tangible benefits for many people. There would be no going to work or school in the dark, and in summer the longer daylight time would automatically come after work/school.
This needs to assume people have decent access to interactive computing power because it would complicate planning things that involve interactions between LZT and World Time, especially when planning events weeks or months ahead.
(And yes, under this system 6:00 AM LZT from one day to 6:00 AM LZT the next day would not be exactly 24 hours. So? Things for which that matters would be scheduled on World Time, not on LZT).