You may have to be prepared for the fact that no records will exist. According to Google Maps, Wallasey is near Liverpool, is it possible that he came in there? Certainly no passenger lists exist for ships originating in the Mediterranean and I suspect this is true for crew lists also. Have you any idea how long the couple were together? If they were together for months/years rather then weeks, he might be listed on local electoral rolls, telephone directories etc. Otherwise, with the thousands of ships using Liverpool and the docks in the surrounding area, I fear you might have a long and potentially unrewarding search. In the conversation where you found out the name, did the person know anything else, a ship name for example? That might help to narrow things a little.
Good luck with your search! :-)