Inherently nonfinitely based additively idempotent semirings

We give a sufficient condition for an additively idempotent semiring to be inherently nonfinitely based. Namely, if its generated variety is locally finite and every Zimin word is minimal in the additive order, then it is contained in no finitely based locally finite variety. For each positive integer $n$, we construct an infinite finitely generated flat semiring satisfying all identities in at most $n$ variables of every semiring with this minimality property. We also characterize Zimin minimality by membership of a countable flat factor semiring $\text{Finf}$ in the generated variety. The variety $\text{V}(\text{Finf})$ is locally finite and inherently nonfinitely based.