  We present an embedded-cluster method, based on the TRILEX formalism. It
turns the Fierz ambiguity, inherent to approaches based on a bosonic decoupling
of local fermionic interactions, into a convergence criterion. It is based on
the approximation of the three-leg vertex by a coarse-grained vertex computed
by solving a self-consistently determined multi-site effective impurity model.
The computed self-energies are, by construction, continuous functions of
momentum. We show that, in three interaction and doping regimes of parameters
of the two-dimensional Hubbard model, self-energies obtained with clusters of
size four only are very close to numerically exact benchmark results. We show
that the Fierz parameter, which parametrizes the freedom in the
Hubbard-Stratonovich decoupling, can be used as a quality control parameter. By
contrast, the $GW$+extended dynamical mean field theory approximation with four
cluster sites is shown to yield good results only in the weak-coupling regime
and for a particular decoupling. Finally, we show that the vertex has spatially
nonlocal components only at low Matsubara frequencies.
