  In a recent contribution we identified possible points of contact between the
asymptotically safe and canonical approach to quantum gravity. The idea is to
start from the reduced phase space (often called relational) formulation of
canonical quantum gravity which provides a reduced (or physical) Hamiltonian
for the true (observable) degrees of freedom. The resulting reduced phase space
is then canonically quantised and one can construct the generating functional
of time ordered Wightman (i.e. Feynman) or Schwinger distributions respectively
from the corresponding time translation unitary group or contraction semigroup
respectively as a path integral. For the unitary choice that path integral can
be rewritten in terms of the Lorentzian Einstein Hilbert action plus observable
matter action and a ghost action. The ghost action depends on the Hilbert space
representation chosen for the canonical quantisation and a reduction term that
encodes the reduction of the full phase space to the phase space of
observavbles. This path integral can then be treated with the methods of
asymptically safe quantum gravity in its {\it Lorentzian} version.
  We also exemplified the procedure using a concrete, minimalistic example
namely Einstein-Klein-Gordon theory with as many neutral and massless scalar
fields as there are spacetime dimensions. However, no explicit calculations
were performed. In this paper we fill in the missing steps. Particular care is
needed due to the necessary switch to Lorentzian signature which has strong
impact on the convergence of ``heat'' kernel time integrals in the heat kernel
expansion of the trace involved in the Wetterich equation and which requires
different cut-off functions than in the Euclidian version. As usual we truncate
at relatively low order and derive and solve the resulting flow equations in
that approximation.
