  We present a semiclassical theory for the scattering matrix ${\cal S}$ of a
chaotic ballistic cavity at finite Ehrenfest time. Using a phase-space
representation coupled with a multi-bounce expansion, we show how the Liouville
conservation of phase-space volume decomposes ${\cal S}$ as ${\cal S}={\cal
S}^{\rm cl} \oplus {\cal S}^{\rm qm}$. The short-time, classical contribution
${\cal S}^{\rm cl}$ generates deterministic transmission eigenvalues T=0 or 1,
while quantum ergodicity is recovered within the subspace corresponding to the
long-time, stochastic contribution ${\cal S}^{\rm qm}$. This provides a
microscopic foundation for the two-phase fluid model, in which the cavity acts
like a classical and a quantum cavity in parallel, and explains recent
numerical data showing the breakdown of universality in quantum chaotic
transport in the deep semiclassical limit. We show that the Fano factor of the
shot-noise power vanishes in this limit, while weak localization remains
universal.
