[Objective]

Originally Posted by Answer

On a fundamental level, is it really possible for the belief of such things to be consistent with the ideas of Atheism?

The words ''atheism'' and ''belief'' can not co-exist. They may claim they're atheists, but they're not. Unless what they've experienced can be explained by science there's no viable reason to believe they're anything else than agnostic. This comes from someone who has been in their shoes.

Atheism and ''no scientific basis other than an unexplained observation'' got nothing in common. I consider myself an agnostic that moves heavily towards atheism, but I can't under any circumstances call myself one as long as I have doubts about my own observations not coinciding with science.
If I were a true atheist I would disregard all of my observations and experiences with what many people explains as ''ghosts'' or ''spirits''. I believe they're tricks by the mind, but since I have slight doubts to that I'm no atheist although I'd like to say that I am one.

If it can be proved by science and endless tests reaching the state of a theory that's backing up their thoughts about it actually being ghosts and spirits they can go back to calling themselves atheists again.