Without seeing the .45 in question and the markings, I'll have to hazard a guess based on some previous experience.

 

John M. Browning had a very close relationship with Fabrique Nationale (F.N.) in Herstal, Belgium, that continued after his death.  One of the marketing arrangements was that F.N. would market his designs under their logo in various areas of the world and under the Browning name in North America.  That's why you'll find Browning products marked as made in Belgium but bearing a Browning Arms logo on the gun.

 

One of the problems for arms manufacturers in general and European in particular is their costs of production in the various countries.  Some Browning High Power 9mm pistols are marked as Brownings, but made by F.N. in Belgium, and assembled in Portugal (where labor costs are presumably lower).

 

Browning marketed a double action pistol called the BDA that was made by SIG Sauer (Switzerland) and carried the Browning name (and the SIG Sauer name).  This maybe what you've discovered by the dual marking on the slide of this pistol.  The key whould be what brand the pistol was sold under; that is, was it sold as a SIG Sauer or as a Browning?  My guess is that its a SIG Sauer design that's sold by Browning Arms in this country.

 

Any other folks out there have any ideas or inputs?