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Short THz pulses are highly attractive both for fundamental research and practical applications in this underdeveloped region of the electromagnetic spectrum. Typically, THz pulses are generated using nonlinear optical methods, starting from a high-power visible or near-IR mode-locked source, but usually suffer from low conversion efficiencies. Here, we present a direct on-chip THz source of coherent short pulse trains based on active mode-locking of an inverse-designed planarized Y-coupled THz quantum cascade laser. By employing quasi-resonant microwave modulation of the asymmetric laser cavity, mode-locked pulses as short as 2.3 ps are generated, with emission bandwidths spanning 500 and 700 GHz at a central frequency of 2.9 THz. Analytical models and numerical simulations reveal that the two crucial aspects of the short pulse generation in this system are the splitting of light into two arms for reduced gain saturation and the asymmetry-induced quasi-continuum of possible lasing modes, effectively acting as a dispersion-compensated cavity.