Folks on a Carnival cruise ship near the Bahamas watched in horror as a search and rescue mission launched for a man who jumped off the ship ... and we've got photos and video from the scene.

A married couple traveling on the Carnival Liberty tells TMZ ... they were sitting at a bar when they heard the "man overboard" call ... and then a rescue boat went into the water, and the captain came on and told passengers they were turning back to look for the person.

We're told the rescue boat ultimately returned to the ship and passengers could see Carnival personnel performing CPR on the man.

Unfortunately, the man did not survive ... and folks on board say word quickly spread.

A Carnival spokesperson tells TMZ ... the guest climbed over his stateroom balcony jumped while the ship was sailing from Celebration Key to Nassau.

Folks on the ship claim the man hit a lifeboat as he fell, before hitting the water ... and his wife was allegedly in the room and screamed, alerting guests in the neighboring cabin, who called for help.