Not to my knowledge, but since Ionia has drawn lots of inspiration from Asia I can imagine them being like flying swords or other immortal treasures.
The new Irelia doesn't seem to have that anymore since it does not fit into her lore, but she may "reconnect" (not really) the shards in order to surf on them like on a flying sword.
Karma on the other hand...she is just a whole different case of being something that is always reincarnated and worshipped. Hers may be part of something that belongs to Karma herself, and not the girl she currently inhabits or even Ionia.
Hope I could help. :)