We investigate the interference of Laughlin quasiparticles (QPs) in the fractional quantum Hall regime that are stochastically injected into a Fabry-Perot interferometer. We find that the effective Aharonov-Bohm (AB) phase accumulated along the interferometer loop acquires an additional contribution of $sin(2\pi \lambda )/2$ per QP present on it, where $\pi \lambda$ is the QP exchange phase. This contribution originates from time-domain braiding processes associated with injected QPs passing the interferometer quantum point contacts. In the limit of symmetric QP injection, the tunneling current noise exhibits AB oscillations as a function of the total injected current, providing access to the exchange phase $\pi \lambda$. In the regime of large total injection, we identify a universal Fano factor that displays power-law scaling and a characteristic phase shift reflecting time-domain QP braiding at the interferometer QPCs. These results are relevant for accessing anyonic exchange statistics in mesoscopic interferometers.