Even if supersymmetric particles are found at the Large Hadron Collider (LHC), it will be difficult to prove that they constitute the bulk of the dark matter (DM) in the Universe using LHC data alone. We study the complementarity of LHC and DM indirect searches, working out explicitly the reconstruction of the DM properties for a specific benchmark model in the coannihilation region of a 24-parameters supersymmetric model. Combining mock high-luminosity LHC data with presentday null searches for gamma rays from dwarf galaxies with the Fermi Large Area Telescope, we show that current Fermi Large Area Telescope limits already have the capability of ruling out a spurious wino-like solution which would survive using LHC data only, thus leading to the correct identification of the cosmological solution. We also demonstrate that upcoming Planck constraints on the reionization history will have a similar constraining power and discuss the impact of a possible detection of gamma rays from DM annihilation in the Draco dwarf galaxy with a Cherenkov-Telescope-Array-like experiment. Our results indicate that indirect searches can be strongly complementary to the LHC in identifying the DM particles, even when astrophysical uncertainties are taken into account.