Abstract For most of cosmic history, the evolution of our Universe has been governed by the physics of a `dark sector', consisting of dark matter and dark energy, whose properties are only understood in a schematic way. The influence of these constituents is mediated exclusively by the force of gravity, meaning that insight into their nature must be gleaned from gravitational phenomena. The advent of gravitational-wave astronomy has revolutionised the field of black hole astrophysics, and opens a new window of discovery for cosmological sources. Relevant examples include topological defects, such as domain walls or cosmic strings, which are remnants of a phase transition. Here we present the first simulations of cosmic structure formation in which the dynamics of the dark sector introduces domain walls as a source of stochastic gravitational waves in the late Universe. We study in detail how the spectrum of gravitational waves is affected by the properties of the model, and extrapolate the results to scales relevant to the recent evidence for a stochastic gravitational wave background. Our relativistic implementation of the field dynamics paves the way for optimal use of the next generation of gravitational experiments to unravel the dark sector.