The Brownian net, roughly speaking, is a collection of branching-coalescing Brownian motions starting from every point in the space-time plane R^2, which has been shown to be the diffusive scaling limit of branching-coalescing simple random walks. The Brownian net is expected to be a universal scaling limit of one-dimensional interacting particle systems with branching-coalescence. However, showing the convergence of models with crossing paths remains a challenge.  We study the model of branching-coalescing nonsimple random walks, where the paths can cross each other. A key ingredient is the duality between the branching-coalescing random walks and the biased voter model. We obtain results for both the Brownian net and the biased voter models.

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