We have constructed relativistic versions of Luttinger fermions in analogy to effective low-energy degrees of freedom of non-relativistic solid-state systems. We propose to use these relativistic versions as fundamental degrees of freedom of interacting quantum field theories. Using the spin-base invariant formalism, we construct the corresponding Clifford algebra and the spin metric for relativistic invariance of the action. The corresponding minimal spinor has 32 complex components, surprisingly matching the degrees of freedom of a standard-model generation including a right-handed neutrino. Owing to their powercounting properties, new asymptotically free self-interacting field theories of relativistic Luttinger fermions can be constructed in d = 4 spacetime dimensions, representing first examples of high-energy complete QFTs based on pure matter degrees of freedom. Gauge theories with relativistic Luttinger fermions exhibit a strong paramagnetic dominance, requiring large nonabelian gauge groups to maintain asymptotic freedom.