Ten years ago V. Fock and A. Goncharov studied moduli spaces of local systems. Certain coordinate systems on moduli spaces of flat connections are induced by spectral networks in a natural way. In the present paper, the authors extend this method by incorporating the complexified Fenchel-Nielson coordinates. The main result here is that abelianization provides a new way of thinking about this classical object. It is briefly reviewed what is meant by the process of nonabelianization. This process is formalized by focusing on the case of \(\mathrm{SL}(2)\)-connections. The paper starts with the definition of a spectral network and introduces various examples. Two special types of spectral network lead to Fock-Goncharov and Fenchel-Nielson coordinate systems. The physical motivation comes from the link between spectral networks and \(\mathcal N=2\) supersymmetric quantum field theories. The present paper gives a new way of understanding the role of Fenchel-Nielson coordinates in these theories. A second interesting consequence of this discussion concerns the physics of line defects. The new point of view is that these defects are Wilson lines for the \(\mathrm{SU}(2)\) gauge symmetry in the fundamental representation.