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We propose to use the DSR eikonal equation for the first-break traveltime tomography. The proposed method relies on an efficient DSR solver, which is realized by a version of the fast-marching method based on an implicitly causal discretization. Since the DSR eikonal equation allows changing of source position along the ray-path, its linearization results in a tomographic inversion that naturally handles possible conflicting information between different shots. Our numerical tests show that, compared to the traditional tomography with a shot-indexed eikonal equation, the DSR tomography converges faster and is more robust. Its benefits may be particularly significant in the presence of noise in the data.