  The primary objective of this study is to investigate hadronic molecules of
$K^*\bar K_1(1270)$ using a one-boson-exchange model, which incorporates
exchanges of vector and pseudoscalar mesons in the $t$-channel, as well as the
pion exchange in the $u$-channel. Additionally, careful consideration is given
to the three-body effects resulting from the on-shell pion originating from
$K_1(1270)\to K^*\pi$. Then the BESIII data of the $J/\psi\to\phi\eta\eta'$
process is fitted using the $K^*\bar K_1(1270)$ scattering amplitude with
$J^{PC}=0^{--}$ or $1^{--}$. The analysis reveals that both the $J^{PC}=0^{--}$
and $1^{--}$ assumptions for $K^*\bar K_1(1270)$ scattering provide good
descriptions of the data, with similar fit qualities. Notably, the parameters
obtained from the best fits indicate the existence of $K^*\bar K_1(1270)$ bound
states, denoted by $\phi(2100)$ and $\phi_0(2100)$ for the $1^{--}$ and
$0^{--}$ states, respectively. The current experimental data, including the
$\eta$ polar angular distribution, cannot distinguish which $K^*\bar K_1(1270)$
bound state contributes to the $J/\psi\to\phi\eta\eta'$ process, or if both are
involved. Therefore, we propose further explorations of this process, as well
as other processes, in upcoming experiments with many more $J/\psi$ events to
disentangle the different possibilities.
