When she thought he was Dyle's brother, she asked him his first name.
Before he answered, he disappeared.
But THREE TIMES afterward, after still not having gotten an answer, she calls him "Alex".
How'd she know his name's "Alex"?
And why, NOT ONCE, did he say: "How'd you know my name's Alex, hmm?"
B .... I .... L-L-T .... E-T-L-E-YYYYYYY Bill Tetley!
Watch the movie again. True, Reggie is left without an answer to her "first name" question when he disappears at the American Express office. When he reappears at their little hotel with a bloody back, she doesn't call him by any name until she remarks (while tending to his wound): "You could at least tell me what your first name is these days." He answers "Alexander," and she responds "OK, Alexander - you're done." She switches to the more familiar "Alex" a few moments later.
It's in the film clip linked below -
http://www.youtube.com/watch?v=NEYwJbjyF2M