there are many ideas on reincarnation, and it is not monopolized by the Eastern Religions this concept of Reincarnation because St. Augustine himself pointed out the idea of Rebirth, the greeks called it metempsychosis.
In Vedantic reincarnation there is no many consciousness, there is only One consciousness but because of ignorance these persons perceive that there are many things, that there is self and there is others, but in reality, there is only One, and this One this single entity is ineffable.
ANd yes Orthodox Chrisitanity doesnt hold the idea of Reincarnation.