Tracking how an infectious disease spreads in time and space relies on several distinct sources of surveillance data, reported case counts, viral concentrations in wastewater, seroprevalence surveys, and pathogen genomic sequences, each of which is imperfect and captures only part of the underlying transmission process. These data streams are typically analyzed separately or with highly parameterized, disease-specific models, making it difficult to combine their complementary strengths. Here we present MASCOT-DataStreams (MASCOT-DS), a BEAST2 software package that extends the structured coalescent model MASCOT to jointly infer prevalence over time and transmission rates between locations from any combination of case counts, wastewater concentrations, seroprevalence surveys, and pathogen phylogenies. Using simulated outbreaks in structured populations, we show that MASCOT-DS accurately recovers true prevalence trajectories and between-location migration rates. We then apply MASCOT-DS to genomic, case count, wastewater, and seroprevalence data from the SARS-CoV-2 Epsilon wave (winter 2020-21) in three San Francisco Bay Area counties, reconstructing county-level prevalence dynamics and quantifying transmission within and into the region. By systematically removing individual data streams, we find that genomic data are uniquely required to estimate transmission between locations, while seroprevalence data are essential for anchoring the overall magnitude of an outbreak; case counts and wastewater concentrations play largely interchangeable roles in capturing outbreak shape. These results demonstrate that integrating complementary epidemiological data streams substantially increases the certainty of transmission dynamics estimates compared to relying on any single data stream, and provides a framework for evaluating the added value of different surveillance strategies.