  We begin this work calculating Halliwell's propagator in the case of a
Mixmaster universe with small anisotropy. Afterwards in the context of the
Decoherence Model we introduce in our system terms that comprise the
self-measurement of the universe of this model by higher multipoles of matter.
Analyzing self-measurement with the Restricted Path Integral Formalism we
obtain Halliwell's modified propagator and find that a gauge invariant physical
time emerges as consequence of this process. The conditions leading to
Wheeler-DeWitt dynamics are also obtained. The comparison of our results with
those of the isotropic case will enable us to conclude that the number of
conditions to be satisfied in order to have Halliwell's regime is in the
anisotropic situation bigger than in an isotropic universe. We obtain also in
terms of the parameters of the measurement process an expression for the
threshold in time beyond which the scale factors of this model are meaningless.
