  Much of our understanding of critical phenomena is based on the notion of
Renormalization Group (RG), but the actual determination of its fixed points is
usually based on approximations and truncations, and predictions of physical
quantities are often of limited accuracy. The RG fixed points can be however
given a fully rigorous and non-perturbative characterization, and this is what
is presented here in a model of symplectic fermions with a nonlocal
("long-range") kinetic term depending on a parameter $\varepsilon$ and a
quartic interaction. We identify the Banach space of interactions, which the
fixed point belongs to, and we determine it via a convergent approximation
scheme. The Banach space is not limited to relevant interactions, but it
contains all possible irrelevant terms with short-ranged kernels, decaying like
a stretched exponential at large distances. As the model shares a number of
features in common with $\phi^4$ or Ising models, the result can be used as a
benchmark to test the validity of truncations and approximations in RG studies.
The analysis is based on results coming from Constructive RG to which we
provide a tutorial and self-contained introduction. In addition, we prove that
the fixed point is analytic in $\varepsilon$, a somewhat surprising fact
relying on the fermionic nature of the problem.
