We report the first implementation of real-time time-dependent density functional theory (RT-TDDFT) at the relativistic four-component level of theory. In contrast to the perturbative linear-response TDDFT approach (LR-TDDFT), the RT-TDDFT approach performs an explicit time propagation of the Dirac-Kohn-Sham density matrix, offering the possibility to simulate molecular spectroscopies involving strong electromagnetic fields while, at the same time, treating relativistic scalar and spin-orbit corrections variationally. The implementation is based on the matrix representation of the Dirac-Coulomb Hamiltonian in the basis of restricted kinetically balanced Gaussian-type functions, exploiting the noncollinear Kramers unrestricted formalism implemented in the program ReSpect. We also present an analytic form for the delta-type impulse commonly used in RT-TDDFT calculations, as well as a dipole-weighted transition matrix analysis, facilitating the interpretation of spectral transitions in terms of ground-state molecular orbitals. The possibilities offered by the methodology are illustrated by investigating vertical excitation energies and oscillator strengths for ground-state to excited-state transitions in the Group 12 atoms and in heavy-element hydrides. The accuracy of the method is assessed by comparing the excitation energies obtained with earlier relativistic linear response TDDFT results and available experimental data.