  In the paper: Fans in the Theory of Real Semigroups. I. Algebraic Theory
(submitted) we introduced the notion of fan in the categories of real semigoups
and their dual abstract real spectra and developed the algebraic theory of
these structures. In this paper we develop the combinatorial theory of
ARS-fans, i.e., fans in the dual category of abstract real spectra. Every ARS
is a spectral space and hence carries a natural partial order called the {\it
specialization partial order}. Our main result shows that the isomorphism type
of a finite fan in the category ARS is entirely determined by its order of
specialization. The main tools used to prove this result are: (1) Crucial use
of the theory of {\it ternary semigroups}, a class of semigroups underlying
that of RSs; (2) Every ARS-fan is a disjoint union of abstract order spaces
(called {\it levels}); (3) Every level carries a natural involution of abstract
order spaces, and (4) The notion of a {\it standard generating system}, a
combinatorial tool replacing, in the context of ARSs, the (absent) tools of
combinatorial geometry (matroid theory) employed in the cases of fields and of
abstract order spaces.
