The classical equations of motion of quantised gauge theories

The Einstein and Maxwell equations are the jewels in the crown of classical physics. But classical physics is only an approximation to nature, arising as a limit of the underlying quantum mechanical description. And in the case of both general relativity and electromagnetism, owing to their gauge theory nature, the full set of classical equations of motion are not guaranteed to follow from the quantum theory. The time-time and time-space components of the Einstein equations in GR and Gauss’ law in EM are enforced ‘by hand' in the quantisation procedure—a choice so as to make the classical-like states behave as per our classical belief.
But what if our universe was actually described by another classical-like state? For GR, the resulting modification of the Einstein equations can be packaged as the inclusion of an auxiliary energy-momentum tensor describing a ’shadow’ matter that adds no additional degrees of freedom to the theory. The homogeneous and isotropic background piece of this auxiliary matter contributes to expansion of the universe identical to cold dark matter, and the inhomogeneous components source curvature perturbations that grow linearly at linear order.