  By investigating the soft gluon radiation in the $J/\psi$ plus jet
photoproduction at the electron-ion collider (EIC), we demonstrate that the
azimuthal angular correlations between the leading jet and heavy quarkonium
provide a unique probe to the production mechanism of the latter. In
particular, a significant $\cos(\phi)$ asymmetry is found for the color-singlet
channel, whereas it vanishes or has an opposite sign for color-octet
production, depending on the jet transverse momentum. Numerical results of
$\cos(\phi)$ and $\cos(2\phi)$ asymmetries employing both the color-singlet
model and the nonrelativistic QCD approach are presented for typical kinematics
at the future EIC.
