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Never the Twain Shall Meet Quest Part 1: Battles at Sea
Talk to Baderon to get the next FFXIV main scenario quest started.
Head down into the hold of the ship and. If you talk to Hob, you will be given the option to go back to Limsa Lominsa; don't do it. Instead head further in to the hold and locate another set of doors.
Approaching the doors will bring up a dialog, asking if you want to leave the area. Say yes and start the next cutscene. Head back out of the hold and say yes to the duty calls dialog.
A cutscene begins and you enter an instance. Emerick and Merodaulyn are fighting on deck; choose which of them to join with and help defeat the other. I personally chose to kill Emerick, since in the previous quests he'd been a real murderous jerk. Unfortunately, this does not actually cause Emerick's death.
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Never the Twain Shall Meet Quest Part 2: Enjoy the Fresh Air and Exercise
Go back to talk to Baderon at the Adventurers' Guild.
Talk to Baderon via the NPC linkpearl; he sends you to the Musketeers' Guild. Teleport back to Limsa Lominsa and head there. Just run in the front doors to get a cutscene going.
Sadly, it's time to run back the far reaches of lower La Noscea, where you just saw SiSipu. As you approach a cutscene will start.
The cutscene ends and an instance begins. Talk to Y'shtola, then walk forward until a new cutscene begins. Be careful to not leave the instance. This cutscene is also wonderful to watch, though not quite as long and intense as the first one.
You wake up to Baderon's cheerful smile; after a short cutscene he gives you your reward and the quest is done.