A little late with this, as have been busy.
A few notes: This won't help everyone, it's legality of use is a bit dubious (I test for bugs and exploits all the time, so... yeah. It could get changed, stopped, this post edited/deleted, etc, but I hope not as no one likes being screwed by RNG and it's not a bug/exploit for free, so I hope SSG sees fit to leave this post be), you still need to farm for them (but now not just on the character that needs them), you'll need the Shared Bank, and it'll only be useful for the Tier 1 shard (Shard of Power). It won't help with Shard of Great Power or Supreme Power.
Process:
Farm for your Shard of Power (from Shroud 1) as normal. If you happen to get it drop for the character you need. Grats.
If on the other hand you've ransacked the chest, don't despair. Get another toon in and continue farming with them for the Shard of Power.
Once you have one, head to the Cannith Crafting station and prep it by Disjuncting it. This will allow you to Cannith Craft on the Shard of Power Trinket and change its status from BTC to BTA.
Place this now BTA Shard of Power Trinket into the Shared Bank so the character you need it on can pick it up.
Craft Green Steel attributes onto the Shard of Power Trinket as your normally would do and use.
If you decide you don't need it afterwards for Green Steel Crafting, you can continue using it for normal Cannith Crafting. There's no wastage here.
Hope this will be able to help some players who are stuck farming. It really does suck when you're at the mercy of RNG.
And I hope the Devs sees fit to make them easier to obtain if they consider this an exploit/bug/Not WAI and needs to be altered so it doesn't work like this. Or add the Great and Supreme versions to Cannith Disjuncting to allow farming on more than just the toon you need. It's bad enough that you need RNG for it to drop, it's even worse when you need to do it on the specific character too.
J1NG
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Example image of process in question:
