The measurement of the intrinsic shape and orientation of dark matter (DM) and intracluster (IC) gas in galaxy clusters is crucial for constraining their formation and evolution, and for enhancing the use of clusters as more precise cosmological probes. Extending our previous works, for the first time we present the results from a triaxial joint analysis of the galaxy cluster Abell 1835, using X-ray, strong lensing (SL) and Sunyaev–Zel'dovich (SZ) data. We parametrically reconstruct the full three-dimensional structure (triaxial shape and principal axis orientation) of both the DM and the IC gas, and the level of non-thermal pressure of the IC gas. We find that the intermediate–major and minor–major axial ratios of the DM are 0.71 ± 0.08 and 0.59 ± 0.05, respectively, and that the major axis of the DM halo is inclined with respect to the line of sight at 18.3 ± 5.2 deg. We present the first observational measurement of the non-thermal pressure out to R₂₀₀. This has been evaluated to be a few per cent of the total energy budget in the internal regions, while it reaches approximately 20 per cent in the outer volumes. We discuss the implications of our method for the viability of the cold dark matter (CDM) scenario, focusing on the concentration parameter C and the inner slope of the DM γ in order to test the CDM paradigm for structure formation. We measure γ = 1.01 ± 0.06 and C = 4.32 ± 0.44; these values are close to the predictions of the CDM model. The combination of X-ray/SL data at high spatial resolution, which are capable of resolving the cluster core, with the SZ data, which are more sensitive to the cluster outer volume, allows us to characterize the level and the gradient of the gas entropy distribution and non-thermal pressure out to R₂₀₀. Thus, we break the degeneracy among the physical models describing the thermal history of the intracluster medium.