I have the view that there can be evidence-based spirituality, but no, NOT "religion".
Buddhism is such a thing. Various mediation techniques are the result of thousands of years of experimentation into the mechanisms and workings of a mind (and possibly spirit). Yoga too.
Scientology pretends to be such a thing.
Religious trappings have been appended to both Buddhism and the Vedas, but in pure form, these are largely exercises in a "scientific exploration of the mind and spirit".
But, by its nature and by definition "religion" involves BELIEF in what is UNSEEN (i.e. "faith"). Believing in what is NOT OBSERVED is the opposite of "evidence". The moment "faith" enters the picture, the evidence goes out the window and is disregarded.
Though getting into subtleties of "mental science", faith can be viewed from a higher plateau as "the imaginative creation of what does not yet exist" - but THAT isn't how it manifests in most religions other than certain branches of Buddhism and Hinduism. Some forms of "prayer" access this factor - though unknowingly.