  Several long, dynamically cold stellar streams have been observed around the
Milky Way Galaxy, presumably formed from the tidal disruption of globular
clusters. In integrable potentials---where all orbits are regular---tidal
debris phase-mixes close to the orbit of the progenitor system. However, the
Milky Way's dark matter halo is expected not to be fully integrable; an
appreciable fraction of orbits will be chaotic. This paper examines the
influence of chaos on the phase-space morphology of cold tidal streams. Streams
even in weakly chaotic regions look very different from those in regular
regions. We find that streams can be sensitive to chaos on a much shorter
time-scale than any standard prediction (from the Lyapunov or
frequency-diffusion times). For example, on a weakly chaotic orbit with a
chaotic timescale predicted to be $>$1000 orbital periods ($>$1000 Gyr), the
resulting stellar stream is, after just a few 10's of orbits, substantially
more diffuse than any formed on a nearby but regular orbit. We find that the
enhanced diffusion of the stream stars can be understood by looking at the
variance in orbital frequencies of orbit ensembles centered around the parent
(progenitor) orbit. Our results suggest that long, cold streams around our
Galaxy must exist only on regular (or very nearly regular) orbits; they
potentially provide a map of the regular regions of the Milky Way potential.
This suggests a promising new direction for the use of tidal streams to
constrain the distribution of dark matter around our Galaxy.
