The supersonic motion of a pulsar in the ambient medium usually renders pulsar wind nebulae with morphologically bow-shaped shocks and/or cometary tails. Chandra, with its unprecedented angular resolution and high sensitivity has proved a great success in the detection and precise characterisation of these fast-moving pulsars. In particular, with an increasing number of detected supersonic pulsar wind nebula systems, an unusual outflow structure misaligned with respect to the pulsar motion has been observed in several cases. Here, we report recent results from the analysis of the extended emission around J1135-6033 and the pulsar itself based on archival Chandra data.