Tidally disrupted globular cluster (GC) 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, which is the tidal debris of a disrupted GC, possesses a secondary diffuse and extended stellar component (∼100 pc wide) around it, detected at the >5 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 GC streams.