  We study the behavior of phase transitions for the four-dimensional charged
Taub-NUT-AdS spacetime with the Newman-Unti-Tamburino (NUT) parameter
interpreted as the thermodynamic multihair in the extended thermodynamic phase
space, and mainly focus on the effects of the NUT parameter on the phase
transitions. We find that there is an upper bound on the value of the NUT
parameter beyond which the corresponding physical inflection point or critical
point will not exist, and the thermodynamic trihair interpretation of the NUT
parameter would admit a little larger upper bound than the thermodynamic bihair
interpretation. Moreover, as long as the NUT parameter is vanishingly small,
the analogy to the van der Waals liquid/gas phase transition is valid
irrespective of the multihair characteristics of the NUT parameter. However, as
the NUT parameter increases to be comparable to the electric charge, such
analogy to the van der Waals system will be broken, and the corresponding
inflection point is not a thermodynamic critical point any more. For a large
NUT parameter, there are frequent occurrences of the zeroth order phase
transition in the case of the thermodynamic bihair interpretation, while only
the first order phase transition happens in the case of the thermodynamic
trihair interpretation.
