  The present paper is a continuation of \cite{jrz} and is devoted to the study
of limit varieties of additively idempotent semirings. A limit variety is a
nonfinitely based variety whose proper subvarieties are all finitely based. We
present concrete constructions for one infinite family of limit additively
idempotent semiring varieties, and one further ad hoc example. Each of these
examples can be generated by a finite flat semiring, with the infinite family
arising by a way of a complete characterisation of limit varieties that can be
generated by the flat extension of a finite group. We also demonstrate the
existence of other examples of limit varieties of additively idempotent
semirings, including one further continuum-sized family, each with no finite
generator, and two further ad hoc examples. While an explicit description of
these latter examples is not given, one of the examples is proved to contain
only trivial flat semirings.
