  In this work we investigate the inclusive photoproduction of the $C$-odd
$S$-wave fully-charmed tetraquark at electron-ion colliders within the
nonrelativistic QCD (NRQCD) factorization framework, at the lowest order in
velocity and $\alpha_s$. The value of the NRQCD long-distance matrix element is
estimated from two phenomenological potential models. Our studies reveal that
the photoproduction of $1^{+-}$ fully-charmed tetraquark may be difficult to
observe at HERA and EicC, nevertheless its observation prospect at EIC appears
to be bright.
