As seen in the Politics SE:
Right now, you can jump on a ferry in France, head over to Cork, drive up to Belfast and catch the ferry to Liverpool, without ever showing your passport to anyone. It seems inconceivable that this arrangement could be maintained if and when the UK leaves the EU.
Best I'm aware you'd need to show your passport before boarding in France, in much the same way that you do when taking the Eurostar to London. Because Ireland, like the UK, isn't part of the Schengen zone.
Which is it?
Bonus question: would you also need to show your passport to go from Belfast to Liverpool, or is that second assertion factual?