You can wear anything you'd like to the shows and no, there is no movie theatre on board.
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Actually, the recommended dress for dinner on any ship with a dress code is meant to apply to the entire evening. However, people often go to shows dressed more casually. As long as you walk directly from your cabin to the theater it is OK, just don't make a habit of walking around the entire ship in your shorts and T-shirt after dinner. People will probably notice. No one will make you get off the ship... but you might get a few looks.
9 times out of 10 people go to the shows either right before or right after dinner, so most folks are still dressed in their dinner clothes, actually the dress code is for the entire evening. Personally if you are wearing some nice comfy casual clothes to dinner why would you want to change into something else and then have to change again to go to bed? that is unless y ou are going swimming. Even on formal nights, most people stay dressed in their formal attire, there are lots of photo opportunites and things going on on formal night that would make you feel a bit out of place if you were in shorts and a t-shirt., just my observation.