Three nearly degenerate pairs of doublet bands are identified in ¹³¹Ba. Two of them, with positive-parity, are interpreted as pseudospin-chiral quartet bands. This is the first time that a complete set of chiral doublet bands built on the pseudospin partners is observed. The chiral bands with opposite parity built on 3-quasiparticle configurations are directly connected by many E1 transitions, without involving an intermediary non-chiral configuration. The observed band structures in ¹³¹Ba have been investigated by using the reflection-asymmetric particle rotor model. The energies and the electromagnetic transition ratios of the three pairs of doublet bands observed in ¹³¹Ba are reproduced and they are interpreted as chiral doublet bands with three-quasiparticle configurations. It is the first time that multiple chiral bands are observed in the presence of enhanced octupole correlations and pseudospin symmetry.