  Tau-lepton decays with up to two $\pi^0$'s in the final state, $\tau^+ \to
\pi^+ \bar{\nu}_\tau$, $\rho^+ (\pi^+\pi^0) \bar{\nu}_\tau$, $a^+_1
(\pi^+\pi^0\pi^0) \bar{\nu}_\tau$, are used to study the performance of the
barrel part of the silicon-tungsten electromagnetic calorimeter (Si-W ECAL) of
the International Large Detector (ILD) at the future $e^+-e^-$ International
Linear Collider. A correct reconstruction of the tau decay mode is crucial for
constraining the tau spin state and measuring the Higgs boson CP state in $H\to
\tau^+\tau^-$ decays. About 95% of $\pi^+ \bar{\nu}_\tau$ and 90% of
$\rho^+\bar{\nu}_\tau$ and $a^+_1\bar{\nu}_\tau$ decays from $e^+e^-\to Z^0\to
\tau^+\tau^-$ reaction at $e^\pm$-beam energy of 125 GeV are correctly
reconstructed. In a smaller ILD detector, with Si-W ECAL radius reduced by
about 20% these numbers degrade by at most 2%. The $\pi^0$ mass resolution
stays below 10%. Since the failures in the tau-lepton reconstruction are mainly
due to photons, the increase of the ILD magnetic field from 3.5 T to 4 T does
not bring any significant improvement.
