  We construct rational all-pass matrix functions with real-valued coefficients
for mirroring pairs of complex-conjugated determinantal roots of a rational
matrix. This problem appears, for example, when proving the spectral
factorization theorem, or, more recently, in the literature on possibly
non-invertible or possibly non-causal vector autoregressive moving average
(VARMA) models. In general, it is not obvious whether the all-pass matrix
function (and as a consequence the all-pass transformed rational matrix with
initally real-valued coefficients) which mirrors complex-conjugated roots at
the unit circle has real-valued coefficients. Naive constructions result in
all-pass functions with complex-valued coefficients which implies that the
real-valued parameter space (usually relevant for estimation) is left.
