  Let X be a smooth projective variety and let G be a connected reductive
group, both defined over a field of characteristic 0. Given a faithful
representation $\rho$ of G into a product of general linear groups, we define a
moduli stack of principal $\rho$-sheaves that compactifies the stack of
G-bundles on X. We apply the theory developed by Alper, Halpern-Leistner and
Heinloth to construct a moduli space of Gieseker semistable principal
$\rho$-sheaves. This provides an intrinsic stack-theoretic construction of the
moduli space of semistable singular principal bundles as constructed by Schmitt
and G\'omez-Langer-Schmitt-Sols.
  Our second main result is the definition of a schematic
Gieseker-Harder-Narasimhan filtration for $\rho$-sheaves, which induces a
stratification of the stack by locally closed substacks. This filtration for a
general reductive group G is a refinement of the canonical slope parabolic
reductions previously considered at the level of points by
Anchouche-Azad-Biswas and as a stratification of the stack by Gurjar-Nitsure.
In an appendix, we apply the same techniques to define
Gieseker-Harder-Narasimhan filtrations in arbitrary characteristic and show
that they induce a stratification of the stack by radicial morphisms.
