The affine plane with doubled origin is formed by gluing two copies of by the identity away from the origin. The overlap is the punctured affine plane, so this scheme is nonseparated and the two-open affine cover is not acyclic for the structure sheaf.
The gluing resembles a non-Hausdorff manifold example, but this node concerns an algebraic scheme and its nonseparatedness, not a real manifold.

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