This is actually pretty annoying...
I've had my character exiled for only a few hours, and I do intend to pick a city eventually... I'm just going through all the bounty quests before I start on the serious faction grinding as they're good AA! However, I noticed I had a lot of Seals of Arad in the shared bank with my Conjy. Logged on the exiled character, went to check some items, noticed there was something I could buy that would be an upgrade.
Went to get the seals from the bank... and I can't. Bit annoying that I now have to wait until I've finished betrayal to get the red adorn slot gauntlets I wanted... but such is life. For the most part, I can survive without the shared bank, but there are some items you can't trade, which means you can't access them at all!