18 USC 44 § 922(i) bans importation of all foreign firearms. 925(d) specifies some exceptions to that ban, mostly sporting firearms. 922(r) further specifies that you cannot assemble any gun from foreign parts that would not be allowed to be imported under 922(i) and 925(d). 27 C.F.R. 478.39 defines what qualifies as a semiautomatic rifle that is assembled from foreign parts.
The CFR states that the rifle cannot contain more than 10 of the following foreign parts:
(1) Frames, receivers, receiver
castings, forgings, or stampings
(2) Barrels
(3) Barrel extensions
(4) Mounting blocks (trunnions)
(5) Muzzle attachments
(6) Bolts
(7) Bolt carriers
(8) Operating rods
(9) Gas pistons
(10) Trigger housings
(11) Triggers
(12) Hammers
(13) Sears
(14) Disconnectors
(15) Buttstocks
(16) Pistol grips
(17) Forearms, handguards
(18) Magazine bodies
(19) Followers
(20) Floorplates
So if your 'post-ban' AK has more than 10 of those parts not made in the US, it is illegal to possess. This is usually referred to as the 922(r) rule.