"But I can't see how science and religion can complement each other. One deals with the natural world and the other deals with a supernatural world or an imaginary world."
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That is exactly why they can complement each other.  They are about different things.  
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Religion can sometimes reflect great psychological and sociological truths - as well as a pile of nonsense.
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But at least as long ago as St. Augustine (if not thousands of years before that), there were religious leaders who sought harmony in nature.  Many early scientists were also clerics - whence came their literacy.
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E.g., Erasmus was one of the founders of modern science.  His more analogical, flexible, approach to biblical interpretation allowed room for science to benefit people generally, as an aid, not a threat.  Luther and Calvin, by contrast, were rather more iconoclastic and dogmatic - not unlike the inquisition.
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Even now, the Pope is a chemist.
There is no necessary contradiction in that.