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 (≈ 30 pc wide), which is the tidal debris of a disrupted globular cluster, possesses a secondary diffuse and extended stellar component (≈ 100 pc wide) around it, detected at > 5σ 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.