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We present X-ray and radio observations of the Fast Blue Optical Transient (FBOT) CRTS-CSS161010 J045834-081803 (CSS161010 hereafter) at t=69-531 days. CSS161010 shows luminous X-ray ($L_x\sim5\times 10^{39}\,\rm{erg\,s^{-1}}$) and radio ($L_ν\sim10^{29}\,\rm{erg\,s^{-1}Hz^{-1}}$) emission. The radio emission peaked at ~100 days post transient explosion and rapidly decayed. We interpret these observations in the context of synchrotron emission from an expanding blastwave. CSS161010 launched a mildly relativistic outflow with velocity $Γβc\ge0.55c$ at ~100 days. This is faster than the non-relativistic AT2018cow ($Γβc\sim0.1c$) and closer to ZTF18abvkwla ($Γβc\ge0.3c$ at 63 days). The inferred initial kinetic energy of CSS161010 ($E_k\gtrsim10^{51}$ erg) is comparable to that of long Gamma Ray Bursts (GRBs), but the ejecta mass that is coupled to the mildly relativistic outflow is significantly larger ($\sim0.01-0.1\,\rm{M_{\odot}}$). This is consistent with the lack of observed gamma-rays. The luminous X-rays were produced by a different emission component to the synchrotron radio emission. CSS161010 is located at ~150 Mpc in a dwarf galaxy with stellar mass $M_{*}\sim10^{7}\,\rm{M_{\odot}}$ and specific star formation rate $sSFR\sim 0.3\,\rm{Gyr^{-1}}$. This mass is among the lowest inferred for host-galaxies of explosive transients from massive stars. Our observations of CSS161010 are consistent with an engine-driven aspherical explosion from a rare evolutionary path of a H-rich stellar progenitor, but we cannot rule out a stellar tidal disruption event on a centrally-located intermediate mass black hole. Regardless of the physical mechanism, CSS161010 establishes the existence of a new class of rare (rate $<0.4\%$ of the local core-collapse supernova rate) H-rich transients that can launch mildly relativistic outflows.