In RuCl3, inelastic neutron scattering and Raman spectroscopy reveal a continuum of non-spin-wave excitations that persists to high temperature, suggesting the presence of a spin liquid state on a honeycomb lattice. In the context of the Kitaev model, finite magnetic fields introduce interactions between the elementary excitations, and thus the effects of high magnetic fields that are comparable to the spin-exchange energy scale must be explored. Here, we report measurements of the magnetotropic coefficient—the thermodynamic coefficient associated with magnetic anisotropy—over a wide range of magnetic fields and temperatures. We find that magnetic field and temperature compete to determine the magnetic response in a way that is independent of the large intrinsic exchange-interaction energy. This emergent scale-invariant magnetic anisotropy provides evidence for a high degree of exchange frustration that favours the formation of a spin liquid state in RuCl3. © 2020, The Author(s), under exclusive licence to Springer Nature Limited.