From Anton Fetisov's work I think we can extract a simple example. In fact this example should be universal for the class of examples he constructs.
Consider the free symmetric monoidal category on one object with duals. We can represent the objects of this category as a pair $(A^+, A^-)$ of finite sets, with a morphism $(A^+, A^-) \to (B^+, B^-)$ being a bijection between $A^+ \circ B^-$ and $A^- \circ B^+$, plus a finite number of loops. Morphisms are composed by gluing the diagrams and following the paths to form a new bijection, while also possibly creating loops.
Suppose we want to add (countably) infinite products by generalizing this to infinite sets. Then we run into trouble, because when we compose the paths, we could get something that goes around infinitely in a spiral rather than ending up at any exit point. But if we just define a morphism $(A^+, A^-) \to (B^+, B^-)$ to be a bijection between a subset of $A^+ \circ B^-$ and a subset of $A^- \circ B^+$, plus a set of loops, then we can compose morphisms as before, throwing out any infinite paths.
This satisfies the axioms of a symmetric monoidal abelian category, with union as the tensor product structure. The element $(\mathbb N,0)$ is idempotent and dual to $(0,\mathbb N)$, but not self-dual as $(\mathbb N,0)$ and $(0, \mathbb N)$ are not isomorphic.