Unfortunately, I believe they would have used "Western" style ships, captured and turned to their own cause. :(
Certainly after a period they stopped (or toned down) the use of galleys, tartanes, feluccas, etc and I'm pretty sure the use of lateen sails is almost confined solely to the Mediterranean only.
Not very exotic I know, but I remember researching this a bit myself when I made my own corsair ship.
(http://i298.photobucket.com/albums/mm243/fsultana/P7140034.jpg)