  Finite-temperature quantum field theories are formulated in terms of Green's
functions and self-energies on the Matsubara axis. In multi-orbital systems,
these quantities are related to positive semidefinite matrix-valued functions
of the Carath\'eodory and Schur class. Analysis, interpretation and evaluation
of derived quantities such as real-frequency response functions requires
analytic continuation of the off-diagonal elements to the real axis. We derive
the criteria under which such functions exist for given Matsubara data and
present an interpolation algorithm that intrinsically respects their
mathematical properties. For small systems with precise Matsubara data, we find
that the continuation exactly recovers all off-diagonal and diagonal elements.
In real-materials systems, we show that the precision of the continuation is
sufficient for the analytic continuation to commute with the Dyson equation,
and we show that the commonly used truncation of off-diagonal self-energy
elements leads to considerable approximation artifacts. Our method paves the
way for the systematic evaluation of Matsubara data with equations of many-body
theory on the real-frequency axis.
