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The fractional quantum Hall states have long been predicted to be a testing ground of fractional (anyonic) exchange statistics. These topological states harbor quasiparticles with fractional charges of both abelian and non-abelian characters. The quasiparticles' charge is commonly determined by shot noise measurements (1, 2), and states' statistics can be revealed by appropriately interfering the quasiparticles. While the multipath Fabry-Perot electronic interferometer (FPI) is easier to fabricate, it is often plagued by Coulomb interactions (3), its area breathes with the magnetic field (4), and its bulk's charges tend to fluctuate (5). Recent FPI experiments employing adequate screening allowed an observation of Aharonov-Bohm (AB) interference at bulk filling $ν$=1/3 (6). In the current work, we chose to employ an interaction-free, two-path, Mach-Zehnder interferometer (MZI), tuned to bulk filling $ν$=2/5. Interfering the outer $ν$=1/3 mode (with the inner $ν$=1/15 mode screening out the bulk), we observed a 'dressed AB' periodicity, with a combined 'bare AB' flux periodicity of three flux-quanta (3$ϕ_0$) and the 'braiding phase' 2$π$/3. This unique interference resulted with an AB periodicity of a single flux-quantum. Moreover, the visibility of the interference, $v_{e/3}$, deviated markedly from that of the electronic one $\it{v}_{e}$, agreeing with the theoretically expected visibility, $\it{v}_{e/3} \sim {\it{v}_e}^3$. With the two non-equivalent drains of the MZI, the fractional visibility peaked away from the ubiquitous transmission-half of the MZI. We provide simple theoretical arguments that support our results. The MZI proves to be a powerful tool that can be used to probe further the statistics of more complex anyonic quasiparticles.