Partially molten rocks (PMR) are characterized by specific and contrasting behaviours. For instance, large-scale and smaller scale structures are consistently oriented in a migmatitic body with those of the surroundings, indicating that the migmatites were deformed as a whole. By contrast, ubiquitous strain partitioning and melt distribution are widely present in the same migmatitic body, reflecting highly heterogeneous strain and intrinsic rheological instabilities. A continuous transition from a liquid-like to a solid-like rheology, as many averaging processes implicitly assume, cannot explain this two-fold information. We develop a full analysis, considering the stress and strain rate, and the relative proportion of melt and solid phases. Temperature varies from T <sub>solidus</sub> to T <sub>liquidus</sub> in a PMR. We also assume that the transition to melting is not dual to crystallization. However, we prefer using the viscosity rather than the stress, since the former is better constrained from experiments. The viscosity of the matrix, which deforms according to a power law, shows shear thinning, whereas that of the melt remains constant. The viscosity contrast between the two phases thus varies with strain rate. The lower the strain rate, the higher is the viscosity contrast, hence instabilities development is controlled by the rheology. The path followed during a transition also controls the intermediate state, and may lead to instabilities, resulting from mechanical reasons or from the respective amount in each phase. In the last case, the concentration in one phase induces instabilities. A surface describing viscosity in a 3D diagram (strain rate-amount of phase-viscosity) is constructed, that presents a cusp shape for low strain rates. The diagram depicts two types of behaviour and a critical state. At high strain, the viscosity contrast between melt and matrix is lowest. The rock behaves as a near-homogeneous body and a continuous description of its rheology may be estimated. Instabilities lead to fabric development resulting from crystals alignment. At low strain rate, three domains are separated by a critical state. When the proportion of one phase is very small, the material behaves as the other end-member. For intermediate proportions, the cusp indicates three possible viscosity values. Two are metastable, whereas the third is virtual. Hence, the viscosity of the mixture jumps back and forth from the viscosity of one phase to that of the other. A similar process occurs for temperature, since the cusp in the viscosity profile has also implications in a diagram linking temperature and stress. Different behaviours result, depending on whether the deformation takes place under a fixed content in each phase, a common stress, a common strain rate or common temperature. We list several implications for partially molten rocks that may explain fabric development, contact melting between crystals, strain localisation, mineral banding, shear heating, welding, stick-slip-like melt extraction, magma fragmentation or formation of strong or fragile glass. A phase diagram that incorporates temperature, stress and concentration is constructed for PMR that bears much similitude with those issued for other soft materials.