  In the setting of a variety $X$ admitting a tilting bundle $T$ we consider
the problem of constructing $X$ as a quiver GIT quotient of the endomorphism
algebra $A=\textrm{End}_X(T)^{\textrm{op}}$ corresponding to the tilting
bundle. We prove that if the tilting equivalence restricts to a bijection
between the skyscraper sheaves of $X$ and the closed points of a quiver GIT
moduli functor for $A$ then $X$ is indeed a fine moduli space for this quiver
GIT moduli functor, and we prove this result without any assumptions on the
singularities of $X$.
  As an application we consider varieties which are projective over an affine
base such that the fibres are of dimension $\le 1$, and the pushforward of the
structure sheaf on $X$ is the structure sheaf on the base. In this situation
there is a particular tilting bundle on $X$ constructed by Van den Bergh, and
our result allows us to reconstruct $X$ as a quiver GIT quotient for an easy to
describe stability condition and dimension vector.
  This result applies to flips and flops in the minimal model program, and in
the situation of flops shows that both a variety and its flop appear as quiver
GIT moduli spaces for algebras produced from different tilting bundles on the
variety. We also give an application to rational surface singularities, showing
that their minimal resolutions can always be constructed as quiver GIT
quotients for specific dimension vectors and stability conditions. This gives a
construction of minimal resolutions as moduli spaces for all rational surface
singularities, generalising the $G$-Hilbert scheme moduli space construction
which exists only for quotient singularities.
