A mysterious X-ray nebula, showing a remarkably linear geometry, was recently discovered close to the Guitar Nebula, the bow-shock nebula associated with B2224+65, which is the fastest pulsar known. The nature of this X-ray feature is unknown, and even its association with pulsar B2224+65 is unclear. We attempt to develop a self-consistent scenario to explain the complex phenomenology of this object. We assume that the highest energy electrons accelerated at the termination shock escape from the bow shock and diffuse into the ambient medium, where they emit synchrotron X-rays. The linear geometry should reflect the plane-parallel geometry of its ambient field. We estimate the Lorentz factor of the X-ray emitting electrons and the strength of the magnetic field. The former (≃108) is close to its maximum possible value, while the latter, at ≃45µG, is higher than typical interstellar values and must have been amplified in some way. The magnetic field must also be turbulent to some degree to trap the electrons sufficiently for synchrotron X-ray emission to occur effectively. We propose a self-consistent scenario in which, by some streaming instability, the electrons themselves generate a turbulent field in which they then diffuse. Some numerical coincidences are explained, and tests are proposed to verify our scenario. Electron leaking may be common in the majority of pulsar bow-shock nebulae, even though the X-ray nebulosity in general is too diffuse to be detectable.