Nope -- not a typo... 356/357 is a discontinued series better known as GM-1800 (357) and GM-1200 (356) from the 70's and had an flying eagle on the box.. It was similar to the 406/407 series. 457 was already around when the manual was written -- Tascam just didn't recommend it.
All the tapes recommended in the 22-2 manual are 1 mil. Keep in mind the 22-2/22-4 are patterned after the X-series consumer decks. Though they have far outdone themselves performance wise they where originally marketed as a poor man's 32. 34, etc.
The whole idea was to give the small studio/home recordist a decent mixdown deck at low cost, using less expensive (at the time) Hi-Fi tape. They do an excellent job. Best tape out there right now is Ampex/Quantegy 407 -- the machine was practically built around that tape.
Other good choices that I've used are Maxell XL1 35-90b, 3M 207 and one standout 1.5 mil tape that gets along real well is AGFA (or BASF/EMTEC) 468. 468 is like John McCain in the Republican Party -- does its own thing, sounds different, feels different, no sticky shed, etc.
I've rambled extensively in past threads about matching tape with machine -- another lost art. I won't repeat it all here except to say that the 1.5-mil tapes like 456, 226, 911 aren't supple enough to conform to factory head contact specs of these "semi-pro" decks. So while the old standby is normally 456, with machines like
the Fostex A-8, R-8, Tascam 22-2, 388 the thinner 407 wins the compatibility test.
457 would be fine too, but its +6 designation is more than the 22-2 can do anyway. It'll sound good, just different than 407 and more expensive. 457 and 456 also weigh a lot more as far as the transport is concerned. Believe it or not, more wear and tear and technically not within wow & flutter specs.