Well, I think the reason why there's little response to this is that very few people have experience with this sort of thing, and extrapolation of "usual" dating experience isn't really possible.
I am no psychologist, psychiatrist or neurologist, (and even if I was, distance diagnosis isn't something a respectable representative of these professions would do), all I can ask you is this: Is the one personality you fancy (and that fancies you) really worth the trouble? And I don't mean trouble as in "mild inconvenience", but as in "life altering changes of monumental proportions". Can you really see yourself spending the rest of your life with someone that is, more or less, two different persons? I know that I couldn't handle such a thing, besides the obvious dangers stemming from a person that is mentally unstable and unpredictable. What if some kind of jealousy or other triangle-affair related problems occur between you and her/them? I know you said they are not aware of each others' actions (an assumption I wouldn't bet on, to be honest... what tells you this isn't going to change in the future, or that the awareness is simply subconcious and influences her actions this way?), but even if they aren't, as soon as another person is involved, the awareness is, sooner or later, going to develope through outside sources. Imagine taking pictures together, or meeting other folks and them later talking about you while her "other her" is active.
I know you cannot reason with emotion, but if I were you, I'd sit down again and think this through very diligently. This isn't a minor mental condition like, say, clinical depression (which in itself can be bad enough), we are talking a major psychological derangement, and one that you can do very little about.
In other words, let me ask the following question: What do you seek to gain from a potential relationship with her?