  Charged lepton system symmetry under combined charge, parity, and
time-reversal transformation (CPT) remains scarcely tested. Despite stringent
quantum-electrodynamic limits, discrepancies in predictions for the
electron-positron bound state (positronium atom) motivate further
investigation, including fundamental symmetry tests. While CPT noninvariance
effects could be manifested in non-vanishing angular correlations between
final-state photons and spin of annihilating positronium, measurements were
previously limited by knowledge of the latter. Here, we demonstrate tomographic
reconstruction techniques applied to three-photon annihilations of
ortho-positronium atoms to estimate their spin polarisation without magnetic
field or polarised positronium source. We use a plastic-scintillator-based
positron-emission-tomography scanner to record ortho-positronium (o-Ps)
annihilations with single-event estimation of o-Ps spin and determine the
complete spectrum of an angular correlation operator sensitive to CPT-violating
effects. We find no violation at the precision level of 10^{-4}, with an over
threefold improvement on the previous measurement.
