Quoted:
Tromix used the build them, from what I can tell, they were the first to do so, or at least offer them commercially.  In 2002 (IIRC) there was a nice article on their .44 Mag version in Gun World.

How it was done:
- the magazines were Desert Eagle mag sleeved in the GI AR15 mags, as someone here already mentioned
- the bolt was a .223 bolt opened up (for the .44 Mag) with custom extractor, for the .357 you should be able to use the 7.62x39 version
- the barrel is the tricky part
–– using the bolt recess as reference, the barrel end that will enter the extension is cut.  Rather than the normal flat face with slight bevel/chamfer toward the chamber, in this case, the chamber has a small lip surrounding it.  This lip has the diameter of the rim and will serve as the resting place for the rim.  This lip PLUS rim will be inside the bolt face when the rifle is in battery (wish I had my CAD folks here to draw this).  Now part of this lip is milled away to accommodate the extractor, including its path during the rotation (locking/unlocking) of the bolt.  Herein lies the challenge in making this beast feed....
- the extension has the center lug ground down to allow center feeding

Normally your round comes out of the mag and encounters the aforementioned chamfer if not properly aligned with the chamber already.  So the nice pointy small diameter bullet fits nicely in the much larger chamber opening and just in case there is a little "ramp" should things not line up perfectly.

Contrast that with the rimmed cartridge version of the AR where (typically) a very blunt bullet of the same (or slightly smaller) diameter as the chamber is trying not to hit this relatively sharp lip that surrounds the chamber PLUS most rimmed cases are totally straight, which does nothing to help with feeding and extraction....  All the elements that you would NOT want in terms of ensuring smooth feeding are heaped into one design.

Can it be done?  Absolutely.  Will it be easy?  No, but then, therein lies the fun and sense of accomplishment, right?

Quoted:
Ron does a lot of amazing things. He feels good about the .357 working
once all of the parts come together...

Dave