  Strong Coulomb interaction in monolayer transition metal dichalcogenides can
facilitate nontrivial many-body effects among excitonic complexes. Many-body
effects like exciton-exciton annihilation (EEA) have been widely explored in
this material system. However, a similar effect for charged excitons (or
trions), that is, trion-trion annihilation (TTA), is expected to be relatively
suppressed due to repulsive like-charges, and has not been hitherto observed in
such layered semiconductors. By a gate-dependent tuning of the spectral overlap
between the trion and the charged biexciton through an "anti-crossing"-like
behaviour in monolayer WS$_2$, here we present an experimental observation of
an anomalous suppression of the trion emission intensity with an increase in
gate voltage. The results strongly correlate with time-resolved measurements,
and are inferred as a direct evidence of a nontrivial TTA resulting from
non-radiative Auger recombination of a bright trion, and the corresponding
energy resonantly promoting a dark trion to a charged biexciton state. The
extracted Auger coefficient for the process is found to be tunable ten-fold
through a gate-dependent tuning of the spectral overlap.
