I will describe joint work with Katzarkov, Noll, and Simpson, which introduces the notion of a versal harmonic map to a building associated with a given spectral cover of a Riemann surface, generalizing to higher rank the leaf space of the foliation defined by a quadratic differential. A motivating goal is to develop a geometric framework for studying spectral networks that affords a new perspective on their role in the theory of Bridgeland stability structures and the WKB theory of differential equations depending on a small parameter. This talk will focus on the WKB aspect: I will discuss the sense in which the asymptotic behavior of the Riemann-Hilbert correspondence is governed by versal harmonic maps to buildings.