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We proceed to study the $CP$ asymmetry in the angular distributions of$\tau\to K_S\pi\nu_\tau$ decays within a general effective field theoryframework including four-fermion operators up to dimension-six. It is foundthat, besides the commonly considered scalar-vector interference, thetensor-scalar interference can also produce a non-zero $CP$ asymmetry in theangular distributions. Bounds on the effective couplings of the non-standardscalar and tensor interactions are obtained under the combined constraints fromthe measured $CP$ asymmetries and the branching ratio of $\tau^-\toK_S\pi^-\nu_\tau$ decay, with $\mathrm{Im}[\hat{\epsilon}_S]=-0.008\pm0.027$and $\mathrm{Im}[\hat{\epsilon}_T]=0.03\pm0.12$, at the scale$\mu_\tau=2~\mathrm{GeV}$ in the $\mathrm{\overline{MS}}$ scheme. Using thebest-fit values, we also find that the distributions of the $CP$ asymmetriescan deviate significantly from the SM expectation in almost the whole $K\pi$invariant-mass region. Nevertheless, the current bounds are still plagued bylarge experimental uncertainties, but will be improved with more precisemeasurements from Belle II as well as the proposed Tera-Z and STCF facilities.Assuming further that the non-standard scalar and tensor interactions originatefrom a weakly-coupled heavy new physics well above the electroweak scale, the$SU(2)_L$ invariance of the resulting SMEFT Lagrangian would indicate that verystrong limits on $\mathrm{Im}[\hat{\epsilon}_S]$ and$\mathrm{Im}[\hat{\epsilon}_T]$ could also be obtained from the neutronelectric dipole moment and the $D^0-\bar{D}^0$ mixing. With the bounds fromthese processes taken into account, it is then found that, unless there existextraordinary cancellations between the new physics contributions, neither thescalar nor the tensor interaction can produce any significant effects on the$CP$ asymmetries in the processes considered.

$CP$ asymmetry in the angular distributions of $\tau\to K_S\pi\nu_\tau$ decays -- II: general effective field theory analysis