  Anomalous contributions to the electric and magnetic dipole moments of the
$\tau$ lepton from new physics scenarios have brought renewed interest in the
development of new charge-parity violating signatures in $\tau$-pair production
at Belle II energies, and also at higher energies of the Large Hadron Collider
and the Future Circular Collider. In this paper, we discuss the effects of spin
correlations, including transverse degrees of freedom, in the $\tau$-pair
production and decay. These studies include calculating analytical formulas,
obtaining numerical results, and building semi-realistic observables sensitive
to the transverse spin correlations induced by the dipole moments of the $\tau$
lepton. The effects of such anomalous contributions to the dipole moments are
introduced on top of precision simulations of $e^-e^+ \to \tau^-\tau^+$,
$q\bar{q} \to \tau^-\tau^+$ and $\gamma\gamma \to \tau^-\tau^+$ processes,
involving multi-body final states. The $\tau$ decays are simulated along with
radiative corrections, in particular electroweak box contributions of $WW$ and
$ZZ$ exchanges are taken into account. Respective extensions of the Standard
Model amplitudes and the reweighting algorithms are implemented into the KKMC
Monte Carlo, which is used to simulate $\tau$-pair production in $e^-e^+$
collisions, and the TauSpinner program, which is used to reweight events with
$\tau$ pair produced in $pp$ collisions.
