We present results from Chandra and XMM - Newton observations of the radio pulsar B1046 - 58 . A high-resolution spatial analysis reveals an asymmetric pulsar wind nebula ( PWN ) \sim 6 ^ { \prime \prime } \times 11 ^ { \prime \prime } in size . The combined emission from the pulsar and its PWN is faint , with a best-fit power-law photon index of \Gamma =1.7 ^ { +0.4 } _ { -0.2 } and unabsorbed luminosity of \sim 10 ^ { 32 } ergs s ^ { -1 } in the 0.5 - 10.0 keV range ( assuming a distance of 2.7 kpc ) . A spatially resolved imaging analysis suggests the presence of softer emission from the pulsar . No pulsations are detected from PSR B1046 - 58 ; assuming a worst-case sinusoidal pulse profile , we derive a 3 \sigma upper limit for the pulsed fraction in the 0.5–10.0 keV range of 53 % . Extended PWN emission is seen within 2 ^ { \prime \prime } of the pulsar ; the additional structures are highly asymmetric and extend predominantly to the south-east . We discuss the emission from the PWN as resulting from material downstream of the wind termination shock , as outflow from the pulsar or as structures confined by a high space velocity . The first two interpretations imply equipartition fields in the observed structures of \gtrsim 40 - 100 \mu G , while the latter case implies a velocity for the pulsar of \gtrsim 190 n _ { 0 } ^ { -1 / 2 } km s ^ { -1 } ( where n _ { 0 } is the ambient number density in units of cm ^ { -3 } ) . No emission from an associated supernova remnant is detected .