Networks are widely used in science and technology to represent relationships between entities, such as social or ecological links between organisms, enzymatic interactions in metabolic systems, or computer infrastructure. Statistical analyses of networks can provide critical insights into the structure, function, dynamics, and evolution of those systems. However, the structures of real-world networks are often not known completely, and they may exhibit considerable variation so that no single network is sufficiently representative of a system. In such situations, researchers may turn to proxy data from related systems, sophisticated methods for network inference, or synthetic networks. Here, we introduce a flexible method for synthesizing realistic ensembles of networks starting from a known network, through a series of mappings that coarsen and later refine the network structure by randomized editing. The method, MUSKETEER, preserves structural properties with minimal bias, including unknown or unspecified features, while introducing realistic variability at multiple scales. Using examples from several domains, we show that MUSKETEER produces the intended stochasticity while achieving greater fidelity across a suite of network properties than do other commonly used network generation algorithms.