When going into ports, I assume there could be some time difference in island time and ship time. How does this work exactly for departing times from the port?
On our last cruise - which was on HAL - the ship told us to change our watches three times for different locations. I'd say pay attention to what the captain tells you. It was also posted in our daily news, so make sure you check that also. If in doubt, ask before you get off the ship.
It is normally the ship's time to stay on for the ports. I too had an experience like Susan mentioned about HAL, except on CCL during a ship repositioning cruise before going in drydock. They had us change our watches for the ports multiple times, and then it was during a EST to DST time change on top of it. No one knew what the real time was including the crew!
The point being, read the daily newsletter and listen to the announcements just in case there is a time change for an odd reason.
Just remember that if you are instructed to stay on ship's time, completely ignore any clocks on shore, and also do not ask any locals for the time. This sounds obvious, yet I've heard stories about pax nearly missing the ship because of it.
This is one time to wear a watch while on vacation.