  We investigate numerically soliton-plasma interaction in a noble-gas-filled
silica hollow-core anti-resonant fiber pumped in the mid-IR at 3.0 {\mu}m. We
observe multiple soliton self-compression stages due to distinct stages where
either the self-focusing or the self-defocusing nonlinearity dominates.
Specifically, the parameters may be tuned so the competing plasma
self-defocusing nonlinearity only dominates over the Kerr self-focusing
nonlinearity around the soliton self-compression stage, where the increasing
peak intensity on the leading pulse edge initiates a competing self-defocusing
plasma nonlinearity acting nonlocally on the trailing edge, effectively
preventing soliton-formation there. As the plasma switches off after the
self-compression stage, self-focusing dominates again, initiating another
soliton self-compression stage in the trailing edge. This process is
accompanied by supercontinuum generation spanning 1-4 {\mu}m. The technique
could be exploited to generate an ultrafast sequence of several few-cycle
pulses.
