  In our previous work, a novel idea that the NUT charge can be thought of as a
thermodynamical multi-hair has been advocated to describe perfectly the
thermodynamical character of the generic four-dimensional Taub-NUT spacetimes.
According to this scheme, the Komar mass M, the gravito-magnetic charge and/or
the dual (magnetic) mass N, together with a new secondary hair J_N=MN, namely,
a Kerr-like conserved angular momentum, enter into the standard forms of the
first law and Bekenstein-Smarr mass formula. Distinguished from other recent
attempts, our consistent thermodynamic differential and integral mass formulae
are both obtainable from a meaningful Christodoulou-Ruffini-type squared mass
formula of almost all of the four-dimensional NUT-charged spacetimes. As an
excellent consequence, the famous Bekenstein-Hawking one-quarter area-entropy
relation can be naturally restored not only in the Lorentzian sector and but
also in the Euclidian counterpart of the generic Taub-NUT-type spacetimes
without imposing any constraint condition. However, only purely
electric-charged cases in the four-dimensional Einstein-Maxwell gravity theory
with a NUT charge have been addressed there. In this paper, we shall follow the
simple, systematic way proposed in that article to further investigate the
dyonic NUT-charged case. It is shown that the standard thermodynamic relations
continue to hold true provided that no new secondary charge is added, however,
the so-obtained electrostatic and magneto-static potentials are not coincident
with those computed via the standard method. To rectify this inconsistence, a
simple strategy is provided by further introducing two additional secondary
hairs: Q_N=QN and P_N=PN, together with their thermodynamical conjugate
potentials, so that the first law and Bekenstein-Smarr mass formula are still
satisfied.
