A clock. No, really, two clocks.
The first, perfectly normal analog 24 hour clock. When it hits 24:00:00, it stops. For 2375 seconds. Then starts again.
The second, probably more complicated, perfectly normal analog 24 hour clock. When it hits 24:00:00, the minute and second hands keep going, but the hour hand stops moving. It doesn't crawl along like normal. When the minute+second hit --:39:34, they swing back to vertical so that what would have been --:39:35 is 00:00:00, and things resume as normal, with the hour hand sweeping slowly along as the minutes move forward.