Pathogen whole-genome sequencing (WGS) has been used to track the transmission of infectious diseases in extraordinary detail, especially for pathogens that undergo fast and steady evolution, as is the case with many RNA viruses. However, for other pathogens evolution is less predictable, making interpretation of these data to inform our understanding of their epidemiology more challenging and the value of densely collected pathogen genome data uncertain. Here, we assess the utility of WGS for one such pathogen, in the "who-infected-whom" identification problem. We study samples from hosts (130 cattle, 111 badgers) with confirmed infection of M. bovis (causing bovine Tuberculosis), which has an estimated clock rate as slow as ~0.1-1 variations per year. For each potential pathway between hosts, we calculate the likelihood that such a transmission event occurred. This is informed by an epidemiological model of transmission, and host life history data. By including WGS data, we shrink the number of plausible pathways significantly, relative to those deemed likely on the basis of life history data alone. Despite our uncertainty relating to the evolution of M. bovis, the WGS data are therefore a valuable adjunct to epidemiological investigations, especially for wildlife species whose life history data are sparse.