Theory: Rand has two souls, that of Rand and LTT. Jordan said uh-uh. That's it.

Theory: Rand has one soul-LTT.

Now, as far as the one-souler rebuttal goes: LTT is a hero of the horn, right? I don't know if that is said straight out in the books but I've always assumed that that was true. A hero of the horn exists in T'A'R till he or she is called out by the horn or woven out by the wheel. Another bit of information we possess is that at least part of the last 3-4 thousand years, Ishy was outside of Shayol Ghul, and would have utilized T'A'R to some extent.
One problem with this. Either LTT or Ishy would almost certainly have sought each other out to resolve their possible conflict once and for all. But that seems not to have happened, since LTT still exists in some form, and so did Ishy, till Rand got him. That says to me that LTT was not in T'A'R. This isn't conclusive, but assume it's true till you get done reading my theory.

Now consider where Rand was born. Dragonmount, where Kinslayer died. RJ doesn't like coincidence, and this is a big one. Unless there is actually a reason that Rand HAD to be born in the same place as LTT. It might be that LTT was stuck on Dragonmount till a chance presented itself. An infant, young enough for LTT to join his soul to, who might not have survived the cold on the mountain top. Rand al'Thor. But a body isn't meant to be possessed. It's obviously not our common state. So when Rand/LTT got to the Two Rivers, LTT may have joined his soul to two others as well. Mat and Perrin. So all three of the ta'veren may be the Dragon Reborn, and all three might be one and a third souls. (Themselves and LTT.)

Most of this is conjecture, something I came up with to fit some of the evidence. But it could explain a great deal. Like the skills of the three ta'veren. Mat is a general, Rand is a channeler, and I think Perrin is meant to be the politician or at least administrator. Now look at LTT. Greatest general of his time, extremely strong in the One Power, and the Tamyrlin. (Sorry, Tamyrlin, but it's true.) Another coincidence?
Then we have the colors in the boys heads when they think of each other. It might be a link between the portions of LTT's soul, one that's finally showing itself to Mat and Perrin. This also might explain something I've been curious about for a while-Mat is definitely mentioned in the prophecies (See CoT, p.616), and might be mentioned another time. Most people know the Seanchans prophecies say the Dragon Reborn will kneel before the Crystal Throne. But that could be Mat instead of Rand, since we all know Rand won't do that voluntarily, but Mat might do it for his wife. And, we have another similariy that might mean quite a bit. Artur Hawkwing noted in The Great Hunt that LTT always got involved with women that got him in trouble. That's as true of Perrin as it is of Rand, and it's also true of Mat.

So there's my theory. I hope it's true.