Yes we all know that any caliber Saiga requires the same requisite number of compliance parts for any given stamped receiver AK. Namely, 5 without muzzle brake, 6 with.
Saiga shotguns have the distinction of being an apparent exception to this mindset. The Saiga 12 shotguns require fewer US compliance parts, according to the original ATF Tech Branch response posted on the (now defunct) SoupBowl website. I managed to find an archive copy of the letter and posted it on my website.
The letters are not as readable as I'd like, and I have not yet had the time to resize them and copy the transcript:
I have uploaded resized and sharpened photos to my gallery here, not sure how readable they are:
The moral of the story is that, according to the tech branch, the Saiga 12 shotgun has only 13 countable parts out of the box! Interesting, no? Apparently the parts are counted differently... I could count them up to 15 parts based on function. That, however, is a discussion for another time.
For the high-cap mag conversion that most people prefer, this means that you need only the mags to meet the "10 or less imported parts" quota. If ever the Tech Branch modifies this parts count, it would be a trivial matter to add more US parts.
Obviously, those who want to create the uber-cool "military" style build will need to check their parts count more closely if they were to use all Russian-built parts. Case in point - most of us will probably use a US-built trigger set simply because they are so nice and easily used. Take that, with the mags or a possible furniture conversion, and it still puts us at < 10 imported parts, even with the additional parts (like muzzle brake, etc).
Let's not forget that our very own Hotbarrel can make pistons for us, so there is another source of US parts.
I have yet to see anything that specifies chokes, etc. as a countable US part, so that one is open for debate.
I am not aware of a rifled barrel for these. Given today's AK smiths know their way around these shotguns, I believe the potential exists for a (US-built!) rifled rebarrel for the slug-only shooter.
Food for thought.