  The search for anyons, quasiparticles with fractional charge and exotic
exchange statistics, has inspired decades of condensed matter research. Quantum
Hall interferometers enable direct observation of the anyon braiding phase via
discrete interference phase jumps when the quasiparticle number changes. Here,
we observe the universal anyonic braiding phase in both the $\nu = 1/3$ and
$4/3$ fractional quantum Hall states by probing three-state random telegraph
noise (RTN) in real-time. We find that the observed RTN stems from anyon
quasiparticle number $n$ fluctuations and reconstruct three Aharonov-Bohm
oscillation signals phase shifted by $2\pi/3$, corresponding to the three
possible interference branches from braiding around $n$ (mod 3) anyons. Hence,
we fully characterize the anyon exchange statistics using new methods that can
readily extend to non-abelian states.
