We show that tropicalization of linear series on curves gives rise to two-parameter families of tilings by polymatroids, with one parameter arising from the theory of divisors on tropical curves and the other from the reduction of linear series of rational functions in non-Archimedean geometry. In order to do this, we introduce a general framework that produces tilings of vector spaces and their subsets by polymatroids. We furthermore show that these tilings are regular and relate them to work by Kapranov and Lafforgue on Chow quotients of Grassmannians.