Tilting theory provides a fascinating link between the representation theory of finite dimensional algebras and algebraic geometry. Traditionally, it is approached from the algebraic geometry side by seeking tilting bundles on projective stacks. However, in studying representation theory, it is much more natural to start with a finite dimensional algebra and ask how one might attempt to construct a projective stack which is derived equivalent to it. In this talk, we look at two moduli stacks which address this question, the moduli of refined representations and tensor stable representations. The key is to incorporate data corresponding to the monoidal structure of the category of coherent sheaves on the derived equivalent stack. This is joint work with Tarig Abdelgadir and Boris Lerner.