Shouldn't they be independent of each other?
Shouldn't all knowledge tie together into a unified whole? Astronomy tells us about stars and planets, physics tells us how they form and thus how the earth formed, chemistry tells us how molecules act and biology tells us about how life behaves.
Why should they be independent?
Our sciences are human constructs, covering what appear to us to be logical subdivisions of natural and cultural history. But those subdivisions are not always clear cut; we have often established arbitrary lines between continuous phenomena.
Given that, there is no need for any of the subfields of science to be independent of all the others. As was pointed out above, "Shouldn't all knowledge tie together into a unified whole?"
What is useful is for individual scientists to act as "peer reviewers" over others in their areas of study, and in closely related areas of study, catching mistakes and suggesting new avenues for research.
Scientists themselves should be independent, but I don't see any way of making all of the subfields of science independent of one another, nor any use in trying to do so.
Religious belief does not constitute scientific evidence, nor does it convey scientific knowledge.
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