I've included as many cards as possible to reduce the number of swaps needed, but some cards might not be as crucial to every deck.

Wanted upgrades:
$$ Necropotence

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Consider adding:
Exsanguinate - Another mana sink that can also be used for value/to stay alive, but Chainer might not have room for another non-creature
Torment of Hailfire - Game ender with enough mana, but Chainer probably doesn't want this
Hagra Mauling - A land that can be used as a removal spell if I have enough lands already
Command Beacon - Phage has it, but the others can also use it even though they don't actually need it
Vesuva, Thespian's Stage - More Cabal Coffers
Deserted Temple (Weathertop)

Noteable exclusions:
Bolas's Citadel - There is no way Chainer can play a lot of cards with this, and Phage probably can't either
Liliana of the Dark Realms - Braids can't really protect it, so it's likely to just be a 4 mana, sorcery speed removal or get a basic to hand
Deadly Rollick - Phage doesn't really want this because it usually wont be free
Phyrexian Arena - While it's always useful, Chainer would rather have creatures
Read the Bones, Night's Whisper, Sign in Blood - Cheap card draw, but Braids doesn't really need them and Chainer would like to avoid them
Dread Presence - Braids doesn't need the card draw, although the life gain is nice and it can even work as a pseudo sac outlet
Nirkana Revenant - 6 mana is a lot, so Braids probably doesn't have time for this
Liliana Vess - Two vampiric tutors, but Phage and Braids will struggle to protect her, and 5 mana for a single, sorcery speed Vamp is too much. Works nice with Bolas's Citadel, though

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Consider cutting from core:
Malakir Rebirth - A recursion/protection spell that can work as a backup land, but Phage might not have the targets and Chainer can just reanimate things
Plaguecrafter - Maybe Phage doesn't need this
Witch's Cottage - Chainer probably doesn't need this
Sidisi, Undead Vizier

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Cuts:
Chainer - This will always be useful, but it's not very on flavor for Phage and Braids doesn't reeeaally need it
Coldsteel Heart - Upgraded to Arcane Signet, although this is probably better than Charcoal Diamond in Chainer because it can be used to get other colors for activated abilities of stolen creatures
Infernal Grasp - Braids and Chainer don't need this, and it has previously been cut from Phage
Swiftfoot Boots - While the protection is very nice, the non-Phage decks don’t really need the haste, and the 1 to equip might actually be a bit more than it's worth sometimes
Mind Stone - The ramp is nice, and so is the option to cycle it, but Chainer doesn’t have a lot of room for non-creatures and not that much use for colourless mana
Manascape Refractor - If I have Coffers already, I'm probably doing OK without this