  There have been extensive recent developments on kagome metals, such as
T$_m$X$_n$ (T= Fe, Co and X= Sn, Ge) and $A$V$_3$Sb$_5$ ($A=$ Cs, K, Rb). An
emerging issue is the nature of correlated phases when topologically
\textit{non-trivial} bands cross the Fermi level. Here, we consider an extended
Hubbard model on the kagome lattice in the presence of spin-orbit couplings,
involving a Kramer's pair of bands that have opposite Chern numbers and are
isolated in the band structure. We construct an effective model in a
time-reversal (T) symmetric lattice description. We determine the correlated
phases of this model and identify a density-wave order in the phase diagram. We
show that this order is T-breaking, which originates from the Wannier orbitals
lacking a common Wannier center -- a fingerprint of the underlying $Z_2$
topology. Implications of our results for the correlation physics of the kagome
metals are discussed.
