The influence of the carrier-envelope phase (CEP) of a pump pulse on the multioctave supercontinuum (SC) generation in a gas-filled anti-resonant hollow-core fiber (AR HCF) by soliton self-compression (SSC) has been explored. We have shown an octave-wide third harmonic generation (THG) in the visible-to-near-infrared range during the pulse compression down to a sub-cycle duration. The CEP of a multi-cycle pump pulse provides control of interference between the third harmonic (TH) and the SC that indicates the coherent synthesis of a sub-cycle pulse with a duration of about 0.4 optical cycles and a peak power of more than 2 GW at the fiber output.