If bosonic particles are cooled down below the temperature of quantum degeneracy, they can spontaneously form a coherent state in which individual matter waves synchronize and combine. Spontaneous coherence of matter waves forms the basis of a number of fundamental phenomena in physics, including superconductivity, superfluidity and Bose-Einstein condensation(1,2). Spontaneous coherence is the key characteristic of condensation in momentum space(3). Excitons-bound pairs of electrons and holes-form amodel system to explore the quantum physics of cold bosons in solids(4,5). Cold exciton gases can be realized in a system of indirect excitons, which can cool down below the temperature of quantum degeneracy owing to their long lifetimes(6). Here we report measurements of spontaneous coherence in a gas of indirect excitons. We found that spontaneous coherence of excitons emerges in the region of the macroscopically ordered exciton state(7) and in the region of vortices of linear polarization. The coherence length in these regions is much larger than in a classical gas, indicating a coherent state with a much narrower than classical exciton distribution in momentum space, characteristic of a condensate. A pattern of extended spontaneous coherence is correlated with a pattern of spontaneous polarization, revealing the properties of a multicomponent coherent state. We also observed phase singularities in the coherent exciton gas. All these phenomena emerge when the exciton gas is cooled below a few kelvin.