  Pulsar wind nebulae (PWNe) produced from supersonic runaway pulsars can
render extended X-ray structures in the form of tails and prominent jets. In
this Letter we report on the analysis of $\sim$130 ks observations of the PWN
around PSR$\,$J1135-6055 obtained with the Chandra satellite. The system
displays bipolar jet-like structures of uncertain origin, a compact nebula
around the pulsar likely formed by the bow shock ahead of it, and a trailing
tail produced by the pulsar fast proper motion. The spectral and morphological
properties of these structures reveal strong similarities with the PWNe in
other runaway pulsars like PSR J1509-5850 and Geminga. We discuss their
physical origin considering both canonical PWN and jet formation models as well
as alternative scenarios that can also yield extended jet-like features
following the escape of high-energy particles into the ambient magnetic field.
