Red Sea rig is a mess dress uniform variant. It is used by some naval officers in the (UK) Royal Navy and other navies, that inherited its traditions (Canadian, Indian, ...). In general, it is similar to usual mess dress, with a jacket replaced by a cummerbund.
Here is an example from a Royal Canadian Navy dress manual:
This concept appeared sometimes in the 19th century. The name indicates that it comes from the Red Sea, probably because it was really uncomfortable to wear a full officer uniform in those parts of the world.
But when did the Red Sea rig appear first? When was it introduced as an official variant? And who introduced it?
Any historical sources on the topic are welcome. Even a painting of an officer in such a dress would be a good clue. Online searches return anything from oil platforms to sail plans, since "rig" is a pretty overloaded term.