I noticed that the MAB gear page on the wiki is rather lacking in all the MAB gear that was possible to get, and didn't include any augmented or synthesis gear. I figured this was a good a place as any to put a comprehensive list of avatar MAB gear and possibly discuss whether replacing certain gear is worth the relativity small amount of +MAB. For example I doubt using the Facio Spats would be worth loosing the TP Bonus from Caller's Spats. I realize there are probably more pieces than I'll be putting on here originally, these are just the ones that I can remember off the top of my head. So, if you have any others, please include any relevant information and I'll edit the post. I'll link these to the wiki for quick reference.
Nirvana [Staff]
How to Obtain: Mythic
MAB:+20(lvl75), +25(lvl80), +30(lvl85), +35(lvl90)
Shareeravadi [Staff]
How to Obtain: Magian Trials
MAB: +10(lvl80), +11(lvl85), +12(lvl90), +13(lvl95)
Soulscourge [Staff]
How to Obtain: NM Hedjedjet
MAB +5
Eidolon Pendant [Neck]
How to Obtain: Walk of Echoes (Vertical Conflux 2)
MAB: +3
Caller's Sash [Waist]
How to Obtain: NM Tonberry Lieje
MAB: +2
Tiresias' Cape [Back]
How to Obtain: NM Halimede
MAB: +1
Esper Stone [Ammo]
How to Obtain: Veridical Conflux 8
MAB: +1
Fervor Ring [Ring]
How to Obtain: The Palborough Project
MAB:~+1 (Ifrit Only)
Diamond Earring [Earring]
How to Obtain: Tenshodo Coffer, Ivory Key, Those Who Lurk in Shadows (III)
MAB: +1~2
Altius Mantle [Back]
How to Obtain: Tenshodo Coffer, Angel skin key, Smash! A Malevolent Menace
MAB: +1~2
Facio Spats [Legs]
How to Obtain: Dominion Tactician, Battle Trophy: 3rd Echelon
MAB: +1~2
Caller's Spats +1/+2 [Legs]
How to Obtain: Magian Trials
TP Bonus: +25%(+1), +50%(+2)