  Tidally disrupted globular cluster streams are usually observed, and
therefore perceived, as narrow, linear and one-dimensional structures in the 6D
phase-space. Here we show that the GD-1 stellar stream ($\approx$ 30 pc wide),
which is the tidal debris of a disrupted globular cluster, possesses a
secondary diffuse and extended stellar component ($\approx$ 100 pc wide) around
it, detected at >5$\sigma$ confidence level. Similar morphological properties
are seen in synthetic streams that are produced from star clusters that are
formed within dark matter sub-halos and then accrete onto a massive host
galaxy. This lends credence to the idea that the progenitor of the highly
retrograde GD-1 stream was originally formed outside of the Milky Way in a now
defunct dark satellite galaxy. We deem that in future studies, this newly found
$cocoon$ component may serve as a structural hallmark to distinguish between
the in-situ and ex-situ (accreted) formed globular cluster streams.
