  L. Louder showed that any generating tuple of a surface group is Nielsen
equivalent to a stabilized standard generating tuple i.e. $(a_1,\ldots
,a_k,1\ldots, 1)$ where $(a_1,\ldots ,a_k)$ is the standard generating tuple.
This implies in particular that irreducible generating tuples, i.e. tuples that
are not Nielsen equivalent to a tuple of the form $(g_1,\ldots ,g_k,1)$, are
minimal. In a previous work the first author generalized Louder's ideas and
showed that all irreducible and non-standard generating tuples of sufficiently
large Fuchsian groups can be represented by so-called almost orbifold covers
endowed with a rigid generating tuple.
  In the present paper a variation of the ideas from \cite{W2} is used to show
that this almost orbifold cover with a rigid generating tuple is unique up to
the appropriate equivalence. It is moreover shown that any such generating
tuple is irreducible. This provides a way to exhibit many Nielsen classes of
non-minimal irreducible generating tuples for Fuchsian groups.
  As an application we show that generating tuples of fundamental groups of
Haken Seifert manifolds corresponding to irreducible horizontal Heegaard
splittings are irreducible.
