The physical properties of magmas play a fundamental role in controlling the eruptive dynamics of volcanoes. Magmas are multiphase mixtures of crystals and gas bubbles suspended in a silicate melt and, to date, no flow laws describe their rheological behaviour. In this study we present a set of equations quantifying the flow of high-viscosity (> 105 Pa·s) silica-rich multiphase magmas, containing both crystals (24–65 vol.%) and gas bubbles (9–12 vol.%). Flow laws were obtained using deformation experiments performed at high temperature (673–1023 K) and pressure (200–250 MPa) over a range of strain-rates (5 · 10− 6 s− 1 to 4 · 10− 3 s− 1), conditions that are relevant for volcanic conduit processes of silica-rich systems ranging from crystal-rich lava domes to crystal-poor obsidian flows. We propose flow laws in which stress exponent, activation energy, and pre-exponential factor depend on a parameter that includes the volume fraction of weak phases (i.e. melt and gas bubbles) present in the magma.