  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 theory
framework including four-fermion operators up to dimension-six. It is found
that, besides the commonly considered scalar-vector interference, the
tensor-scalar interference can also produce a non-zero $CP$ asymmetry in the
angular distributions. Bounds on the effective couplings of the non-standard
scalar and tensor interactions are obtained under the combined constraints from
the measured $CP$ asymmetries and the branching ratio of $\tau^-\to
K_S\pi^-\nu_\tau$ 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 the
best-fit values, we also find that the distributions of the $CP$ asymmetries
can deviate significantly from the SM expectation in almost the whole $K\pi$
invariant-mass region. Nevertheless, the current bounds are still plagued by
large experimental uncertainties, but will be improved with more precise
measurements from Belle II as well as the proposed Tera-Z and STCF facilities.
Assuming further that the non-standard scalar and tensor interactions originate
from a weakly-coupled heavy new physics well above the electroweak scale, the
$SU(2)_L$ invariance of the resulting SMEFT Lagrangian would indicate that very
strong limits on $\mathrm{Im}[\hat{\epsilon}_S]$ and
$\mathrm{Im}[\hat{\epsilon}_T]$ could also be obtained from the neutron
electric dipole moment and the $D^0-\bar{D}^0$ mixing. With the bounds from
these processes taken into account, it is then found that, unless there exist
extraordinary cancellations between the new physics contributions, neither the
scalar nor the tensor interaction can produce any significant effects on the
$CP$ asymmetries in the processes considered.
