We describe a statistical framework for reconstructing the sequence of transmission events between observed cases of an endemic infectious disease using genetic, temporal and spatial information. Previous approaches to reconstructing transmission trees have assumed all infections in the study area originated from a single introduction and that a large fraction of cases were observed. There are as yet no approaches appropriate for endemic situations in which a disease is already well established in a host population and in which there may be multiple origins of infection, or that can enumerate unobserved infections missing from the sample. Our proposed framework addresses these shortcomings, enabling reconstruction of partially observed transmission trees and estimating the number of cases missing from the sample. Analyses of simulated datasets show the method to be accurate in identifying direct transmissions, while introductions and transmissions via one or more unsampled intermediate cases could be identified at high to moderate levels of case detection. When applied to partial genome sequences of rabies virus sampled from an endemic region of South Africa, our method reveals several distinct transmission cycles with little contact between them, and direct transmission over long distances suggesting significant anthropogenic influence in the movement of infected dogs.