From what I understand, a split soul will take you to a specific awakening level from the previous one, e.g. a 5* split soul will let you upgrade the champion from 4* to 5* only. A perfect soul will upgrade the champion to the specified awakening level from any previous one, e.g. a 5* perfect soul will upgrade the champion to 5* from 0-4*.

So if you get a split or perfect soul that has an awakening level less than what your champion already has, then it's essentially useless to you unless you plan on building a duplicate of that champion. If not, you may as well sell it to get some resources back.

I do get the feeling the system is pointlessly complicated though, and they certainly should've come up with some better names for the resources. If we played a drinking game based on whenever the word "soul" was mentioned, we'd all be dead of severe alcohol poisoning before we even got halfway through the explanation for the system.

maybe hellades could give us a tutorial 

"I think my biggest question is -- if I have Awakened a Champion, and now two of six stars are red, and I get another soul with 1 star -- what do I do with it?"

You sell it. It cannot be used for that champ anymore. 

To progress that champ, you have to pull either a 3 star split soul...or a 3 to 6 star perfect soul.

Hope that helps @xonak 

Where do you get these souls??

I have only seen them during the Elhain Quest.  Didnt know what the were for

Iron Twins.