Given a Calabi-Yau complete intersection in a toric variety, certain mirror constructions give different mirrors. Whether they give the same answer depends on if one's complete intersection lies in a special anti-canonical linear system instead of the full linear system. We will explain this phenomenon in the context of homological mirror symmetry. We will focus on studying this phenomenon in the context of some of our favourite examples. This talk is based on joint work with C. Doran and D. Favero (Alberta).