  D.~Gaiotto, G.~W.~Moore and A.~Neitzke introduced spectral networks to
understand the framed $G$-local systems over punctured surfaces for $G$ a split
Lie group via a procedure called abelianization. We generalize this
construction to groups $G$ of the form $\mathrm{GL}_2(A)$, where $A$ is a
unital associative ring, and to some of its subgroups. This relies on a precise
analysis of the degree 2 ramified coverings associated with spectral networks
and triangulations and on a matrix reinterpretation of their path lifting
rules; along the way we provide another proof of the Laurent phenomenon brought
to light by A.~Berenstein and V.~Retakh. The partial abelianization enables us
to gives parametrizations of the moduli spaces of decorated $G$-local systems
and of framed $G$-local systems over punctured surfaces. For $(A, \sigma)$ a
Hermitian involutive $\mathbf{R}$-algebra the group $G=\mathrm{Sp}_2(A,
\sigma)$ is a classical Hermitian Lie group of tube type, and we are able to
identify and parametrize the moduli space of maximal framed $G$-local systems.
