  Our detailed Angle Resolved Photoemission Spectroscopy (ARPES) study of
$2H$-TaS$_2$, a canonical incommensurate charge density wave (CDW) material,
illustrates pronounced many-body renormalization in the system, which is
manifested by the presence of multiple kink structures in the electronic
dispersions. Temperature-dependent measurements reveal that these kink
structures persist even at temperatures higher than the charge density wave
transition temperature $\it{T}_{\text{cdw}}\,$ and the energy locations of the
kinks are practically temperature-independent. Correlating kink energies with
the published Raman scattering data and the theoretically calculated phonon
spectrum of $2H$-TaS$_2$, we conclude phononic mechanism for these kinks. We
have also detected momentum-anisotropy in the band renormalization, which in
turn indicates momentum-dependence of the electron-phonon coupling of the
system.
