  In this paper, we present detailed studies on the feasibility at $pp$, $e^-p$
and $e^+e^-$ colliders for model-independent sensitivity estimates on the total
cross-section and on the anomalous $\tau^+\tau^-\gamma$ interaction through the
tau pair production channels $pp \to p\tau\bar \tau \gamma p$, $e^-p \to e^-
\tau\bar \tau \gamma p$ and $e^+e^- \to e^+ \tau\bar \tau \gamma e^-$ at the
$\gamma^*\gamma^* \to \tau^+\tau^-\gamma$ mode. Measurements of the anomalous
couplings of the $\tau$-lepton $\tilde{a}_\tau$ and $\tilde{d}_\tau$ provide an
excellent opportunity to probing extensions of the Standard Model. We estimate
the sensitivity at the $95\%$ Confidence Level, and we consider that the
$\tau$-lepton decays leptonically or semi-leptonically. We found that of the
three considered colliders, the future CLIC at high energy and high luminosity
should provide the best sensitivity on the dipole moments of the $\tau$-lepton
$\tilde a_\tau= [-0.00128, 0.00105]$ and $ |\tilde{d}_\tau({\rm ecm})|=
6.4394\times 10^{-18}$, which show a potential advantage compared to those from
LHC and FCC-he.
