If it pretends to be a query about what the correct shape is, it is not only out of place in a dictionary but is a trick question : it lacks words. Colloquially they ended up using both, in most cases the least bad is the second (the first would originate from an Italianism); but in a forum with prepared people this could generate an interesting debate. Not here.
BASE A or BASED IN adverbial locution that anticipates the reasons or backgrounds that are held (to perform or because an action was taken or happened or any event will occur). WITH BASE IN is used, in addition and mainly, in the sense of a place of a military installation, an industry or government.
