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I think of Rei more as reincarnating, like Vishnu, than as any kind of trinity- though certainly in Third Impact her three forms seem to be treated as somewhat independent.
To be honest it seems to me that reincarnation is the standard fate of souls in Eva's cosmology- this explains how people are still being born even after Lilith's Chamber of Guf is empty, and renders the transfer of souls between bodies that Nerv and Seele do regularly just a controlled version of a natural process. Moreover reincarnation is, for the Japanese, a default notion of death- just as a pseudo-Christian heaven and hell are for even non-religious Westerners. This makes Seele's motives even clearer- they're essentially materialist Buddhists (or Schopenhauerists).
Rei, of course, has a soul with a pretty high power level, so she retains her individuality a lot more than most do upon reincarnation.
One might also bring up the idea, current on Evageeks, that Rei II does not contain all of Rei's soul, the other part being inside Unit-00. What are y'all's thoughts on that?