Mantle viscosity controls a variety of geodynamic processes such as glacial isostatic adjustment (GIA), but it is poorly constrained because it cannot be measured directly from geophysical measurements. Here we develop a method that calculates viscosity using empirical viscosity flow laws coupled with mantle parameters (temperature and water content) inferred from seismic and magnetotelluric (MT) observations. We find that combining geophysical constraints allows us to place significantly tighter bounds on viscosity estimates compared to using seismic or MT observations alone. In particular, electrical conductivity inferred from MT data can determine whether upper mantle minerals are hydrated, which is important for viscosity reduction. Additionally, we show that rock composition should be considered when estimating viscosity from geophysical data because composition directly affects seismic velocity and electrical conductivity. Therefore, unknown composition increases uncertainty in temperature and water content, and makes viscosity more uncertain. Furthermore, calculations that assume pure thermal control of seismic velocity may misinterpret compositional variations as temperature, producing erroneous interpretations of mantle temperature and viscosity. Stress and grain size also affect the viscosity and its associated uncertainty, particularly via their controls on deformation regime. Dislocation creep is associated with larger viscosity uncertainties than diffusion creep. Overall, mantle viscosity can be estimated best when both seismic and MT data are available and the mantle composition, grain size and stress can be estimated. Collecting additional MT data probably offers the greatest opportunity to improve geodynamic or GIA models that rely on viscosity estimates.