356 & 357 were just 456 & 457 marketed to consumers through hi-fi shops. 456 had already been around a while. As far as I know it was pretty much the same stuff.
It's likely the 350 series was less perfect than what Ampex was supplying to studios as 456. They cut the tape from huge sheets that are over 2 feet wide. The stuff in the middle is "prime cut" and the stuff near the edges is acceptable, but not flawless. There was an 031 designation for rejected 456, but it could be just fine.
I suspect 356/357 was a step above 031, but not close enough to the center for Ampex to risk screwing up a session at Abbey Road or something.
Here is another box design from 1980 for 7" 357 (1.0 mil) that was around in addition to the one with the eagle on it.
