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We study resolutions of the rational map to the moduli space of stable curvesthat associates with a point in the Hitchin base the spectral curve. In therank two case the answer is given in terms of the space of quadraticmulti-scale differentials introduced in [BCGGM3]. This space defines acompactification (of the projectivization) of the regular locus of the$\mathrm{GL}(2,\mathbb{C})$-Hitchin base and provides a compactification of theHitchin system by compactified Jacobians of pointed stable curves. We show howthe classical $\mathrm{GL}(2,\mathbb{C})$- and$\mathrm{SL}(2,\mathbb{C})$-spectral correspondence extend to the compactifiedHitchin system by a correspondence along an admissible cover betweentorsion-free rank $1$ sheaves and (multi-scale) Higgs pairs of rank $2$.

Compactifying the rank two Hitchin system via spectral data on semistable curves