We perform the first self-consistent measurement of the rate of interactions between stellar tidal streams created by disrupting satellites and dark subhalos in a cosmological simulation of a Milky-Way-mass galaxy. Using a retagged version of the Aquarius A dark-matter-only simulation, we selected 18 streams of tagged star particles that appear thin at the present day and followed them from the point their progenitors accrete onto the main halo, recording in each snapshot the characteristics of all dark-matter subhalos passing within several distance thresholds of any tagged star particle in each stream. We considered distance thresholds corresponding to constant impact parameters (1, 2, and 5 kpc), as well as those proportional to the region of influence of each subhalo (one and two times its half-mass radius ). We then measured the age and present-day, phase-unwrapped length of each stream in order to compute the interaction rate in different mass bins and for different thresholds, and compared these to analytic predictions from the literature. We measure a median rate of interactions within 1 (2, 5) kpc of the stream per 10 kpc of stream length per 10 Gyr. Resolution effects (both time and particle number) affect these estimated rates by lowering them.