I am wondering if people have thoughts, tips, suggestions on how to build a subtle world within the world we already live in. I am writing a novel where the world is Earth, no different from our own in any sense, but for the fact that my character is having what others would consider to be super-natural experiences (psychic experiences, experimenting with time travel).  I am trying to strike a balance between making her actions and experiences be anomalous, but at the same time, not wanting to take so much time having to explain , convince, and provide back story on every single anomalous experience.

Two stories I can think of that strike this balance very well are Ted Chiang's story called Story of Your Life and the Ursula K. LeGuin novel, The Lathe of Heaven. In both of these stories, the characters live in an otherwise normal world (arguably in the future, but still consistent with the current rate of technological advancement), but have these amazing, supernatural experiences that, while being anomalous, are somehow acceptable to the world and the other characters in it without too much disbelief.  The authors do not have to spend much time trying to convince either the reader or other characters that these experiences are in fact happening, and can therefore keep the plot moving smoothly.

I hope this makes sense! If anyone has had any experience with building these types of worlds, i would appreciate your insight.