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Small, rocky planets have been found orbiting in extreme proximity to their host stars, sometimes down to only $\sim 2$ stellar radii. These ultra-short-period planets (USPs) likely did not form in their present-day orbits, but rather migrated from larger initial separations. While tides are the probable cause of this migration, the tidal source has remained uncertain. Here we introduce planetary obliquity tides as a natural pathway for the production of USPs within close-in multi-planet systems. The crucial idea is that tidal dissipation generally forces planetary spin vectors to equilibrium configurations called "Cassini states", in which the planetary obliquities (axial tilts) are non-zero. In these cases, sustained tidal dissipation and inward orbital migration are inevitable. Migration then increases the obliquity and strengthens the tides, creating a positive feedback loop. Thus, if a planet's initial semi-major axis is small enough ($a \lesssim 0.05$ AU), it can experience runaway orbital decay, which is stalled at ultra-short orbital periods when the forced obliquity reaches very high values ($\sim 85^{\circ}$) and becomes unstable. We use secular dynamics to outline the parameter space in which the innermost member of a prototypical Kepler multiple-planet system can become a USP. We find that these conditions are consistent with many observed features of USPs, such as period ratios, mutual inclinations, and occurrence rate trends with stellar type. Future detections of stellar obliquities and close-in companions, together with theoretical explorations of the potential for chaotic obliquity dynamics, can help constrain the prevalence of this mechanism.