  Motivated by the study of Polyakov lines in gauge theories, Hanada and
Watanabe recently presented a conjectured formula for the distribution of
eigenphases of Haar-distributed random SU(N) matrices ($\beta$=2), supported by
explicit examples at small N and by numerical samplings at larger N. In this
note, I spell out a concise proof of their formula, and present its orthogonal
and symplectic counterparts, i.e. the eigenphase distributions of Haar-random
unimodular symmetric ($\beta$=1) and selfdual ($\beta$=4) unitary matrices
parametrizing SU(N)/SO(N) and SU(2N)/Sp(2N), respectively.
