A delta-matroid is a generalization of a matroid. It is a set system $(E, \mathcal{F})$ with $\mathcal{F}\neq \emptyset$ satisfying the symmetric exchange axiom:
$$ \forall X, Y \in \mathcal{F},\forall u \in X \triangle Y\ \exists v \in X \triangle Y:X \triangle\{u, v\} \in \mathcal{F} . $$
Every matroid is a delta-matroid, and there are plenty natural set systems in combinatorics that form a matroid (see wikipedia for some standard examples). I am interested in learning about more examples of delta-matroids that are not matroids, but are still somewhat natural set systems (feel free to interpret this loosely). What are some examples? I would be curious to hear about any examples you may know.