  We study the prospects of measuring the CP property of the Higgs ($h$)
coupling to tau leptons using the vector boson fusion (VBF) production mode at
the high-luminosity LHC. Utilizing the previously proposed angle between the
planes spanned by the momentum vectors of the $(\pi^+\pi^0)$ and $(\pi^-
\pi^0)$ pairs originating in $\tau^\pm$ decays as the CP-odd observable, we
perform a detailed Monte Carlo analysis, taking into account the relevant
standard model backgrounds, as well as detector resolution effects. We find
that excluding a pure CP-odd coupling hypothesis requires $\mathcal{O}(400
{~\rm fb}^{-1})$ luminosity at the 14 TeV LHC, and values of the CP-mixing
angle larger than about $25^\circ$ can be excluded at $95\%$ confidence level
using $3 {~\rm ab}^{-1}$ data. It is observed that the uncertainty in the
angular resolution of the neutral pion momenta does not constitute a
significant hurdle. Achieving a signal to background ratio ($S/B$) close to
one, while keeping a high enough signal yield required to study the angular
distributions selects out VBF as a promising mode to probe the CP nature of the
$h\tau\tau$ coupling, with gluon fusion suffering from a low $S/B$, and the
$W^\pm h/Zh$ mode (with leptonically decaying $W^\pm /Z$) having a much smaller
signal rate.
