Understanding the nature and behavior of excitations in quantum spin liquids, and in topological phases of matter in general, is of fundamental importance and has proven crucial for experimental detection and characterization of candidate materials. Current theoretical and numerical techniques, however, have limited capabilities, especially when it comes to studying gapped excitations. Here, we propose a semiclassical numerical method to study systems whose spin liquid behavior is underpinned by perturbative ring-exchange Hamiltonians. Our method can readily access both thermodynamic and spectral properties. We focus in particular on quantum spin ice and its photon and vison excitations. After benchmarking the method against existing results on photons, we use it to characterize visons and their thermodynamic behavior, which remained hitherto largely unexplored. We find that visons, in contrast to spinons in classical spin ice, form a weak electrolyte: Vison pairs are the dominant population at low temperatures. This is reflected in the behavior of thermodynamic quantities, such as pinch point motifs in the relevant correlators. Visons also appear to strongly hybridize with the photon background, a phenomenon that affects the way these quasiparticles may show up in inelastic response measurements. Our results demonstrate that the method, and generalizations thereof, can substantially help our understanding of quasiparticles and their interplay in quantum spin ice and other quantum spin liquids, quantum dimer models, and lattice gauge theories in general.