I just obtained a B1.
The box is a fake for sure (no made in France Bakelite stamp, twin piano hinges not one, logo not wide enough on lid etc) but the outer box seems real. The papers are real I think - too good to be fakes - although the original warranty is missing and so is the COSC certificate.
The watch is either real or the finest fake yet created! It has all the correct appearance, the date code in the right place (4504 B4) on the case, date code on Fighter bracelet, all dial printing correct and very crisp, slide rule works precisely as it should with the gearings etc, flashes the correct misprint as battery end of life warning, has been accurate to spec and is still dead on the second 3 days after setting which I am sure a cheap fake movement would not be. The 'B' is present on the edge of the bezel too - something I doubt fake ones bother with.
It totally feels real and looks it in every way apart from the replaced box and missing original warranty.
Since the watch was made in 2004 it seems perfectly feasible that someone lost their original box, bought what they thought was a new one from Ebay or somewhere and lost the paperwork in a house move or something - or chucked it once the warranty expired etc.
s/n looks real - it is 3461**. Pic attached.
I'm 99% sure the watch is real - as others have said, there are no good fakes of the B1 around and this would have to be a top quality one if it was! I shall find out shortly anyway as it has to go back to Breitling Australia for a battery and maintenance but I attach a pic anyway!