Okay, this is totally subjective but what I take from the song is this - she was with someone that, although she loves them, she knows for whatever reason that they can't be together.
And so as they separate and when they're apart, she thinks about what life could have been and she imagines what life would be like now if they were together. There are probably good reasons why they aren't together and she probably knows that if they were, life wouldn't be anything like what she dreams ("I want you as the dream, not the reality,") and as she sees him with someone else, she knows that it truly is a dream.
Okay, this is totally subjective but what I take from the song is this - she was with someone that, although she loves them, she knows for whatever reason that they can't be together.
And so as they separate and when they're apart, she thinks about what life could have been and she imagines what life would be like now if they were together. There are probably good reasons why they aren't together and she probably knows that if they were, life wouldn't be anything like what she dreams ("I want you as the dream, not the reality,") and as she sees him with someone else, she knows that it truly is a dream.