  We analyze how to parametrize general modifications of the dark matter
perturbations equations in a model-independent way. We prove that a general
model with an imperfect and non-conserved dark matter fluid with bulk and shear
viscosities and heat flux in a modified gravity scenario can be described with
five general functions of time and scale. We focus on the sub-Hubble regime
within the quasi-static approximation and calculate the observable power
spectra of the galaxy distribution, galaxy velocities and weak lensing and find
that these observables are only sensitive to three combinations of the initial
five functions. Deviations of these three observable functions with respect to
$\mathrm{\Lambda CDM}$ give us different characteristic signals which allow us
to determine in which cases it is possible to discriminate a modification of
gravity from an imperfect or non-conserved dark matter. Finally, we perform a
Fisher forecast analysis for these three parameters and show an example for a
particular model with shear viscosity.
