  Generation of topological phases of matter with SU(3) symmetry in a condensed
matter setup is challenging due to the lack of an intrinsic three-fold
chirality of quasiparticles. We uncover two salient ingredients required to
express a three-component lattice Hamiltonian in a SU(3) format with
non-trivial topological invariant. We find that all three SU(3) components must
be entangled via a gauge field, with opposite chirality between any two
components, and there must be band inversions between all {\it three}
components in a given eigenstate. For spinless particles, we show that such
chiral states can be obtained in a tripartite lattice with three inequivalent
lattice sites in which the Bloch phase associated with the nearest neighbor
hopping acts as $k$-space gauge field. The second and a more crucial criterion
is that there must also be an odd-parity Zeeman-like term, i.e.
$\sin(k)\sigma_z$ term where $\sigma_z$ is the third Pauli matrix defined in
any two components of the SU(3) basis. Solving the electron-photon interaction
term in a periodic potential with a modified tight-binding model, we show that
such a term can be engineered with site-selective photon polarization. Such
site selective polarization can be obtained in multiple ways, such as using
Sisyphus cooling technique, polarizer plates, etc. With the $k$-resolved Berry
curvature formalism, we delineate the relationship between the SU(3) chirality,
band inversion, and $k$-space monopoles, governing finite Chern number without
breaking the time-reversal symmetry. The topological phase is affirmed by edge
state calculation, obeying the bulk-boundary correspondence.
