  We present a detailed study, at the fast detector simulation level, of
top-pair photoproduction in semileptonic mode at the LHeC and FCC-he future
colliders. We work in full tree-level QED, not relying on the equivalent-photon
approximation, taking into account the complete photoproduction kinematics.
This allows us to define three photoproduction regions based on the angular
acceptance range of the electron tagger. Those regions provide different
degrees of sensitivity to top-quark effective couplings. We focus on the
$t\bar{t}\gamma$ dipole couplings and the left-handed vector $tbW$ coupling for
which we determine limits at both energies and in different photoproduction
regions. We find that the LHeC and FCC-he will yield tight direct bounds on top
dipole moments, greatly improving on current direct limits from hadron
colliders, and direct limits on the $tbW$ coupling as restrictive as those
expected from the HL-LHC. We also consider indirect limits from $b\to s \gamma$
branching ratio and $CP$ asymmetry, that are well known to be very sensitive
probes of top electromagnetic dipole moments.
