  Using the production reactions $pd\to {}^3\mbox{He}\,\omega$ and $pp\to
pp\omega$, the Dalitz plot distribution for the $\omega \to \pi^+ \pi^- \pi^0$
decay is studied with the WASA detector at COSY, based on a combined data
sample of $ (4.408\pm 0.042) \times 10^4$ events. The Dalitz plot density is
parametrised by a product of the $P$-wave phase space and a polynomial
expansion in the normalised polar Dalitz plot variables $Z$ and $\phi$. For the
first time, a deviation from pure $P$-wave phase space is observed with a
significance of $4.1\sigma$. The deviation is parametrised by a linear term
$1+2\alpha Z$, with $\alpha$ determined to be $+0.147\pm0.036$, consistent with
the expectations of $\rho$-meson-type final-state interactions of the $P$-wave
pion pairs.
