The join of two abstract simplicial complexes $K$ and $L$, denoted $K\star L$ is defined as a simplicial complex on the base set $V(K)\dot{\cup} V(L)$ whose simplices are disjoint union of simplices of $K$ and of $L$ (here $V(K)$ denotes the base set of $K$, which we take to be finite.)

I was wondering if there exist generalization of the join in the following sense:

Suppose $X_1\star X_2 \star \ldots \star X_n $ is the join of finitely many simplicial complexes. If we take one point from each $X_i$ in the join, we obtain a simplex spanned by the points.

Is there an analogous $n$-ary operation $\mathcal{J}(X_1,\ldots , X_n)$ that gives a simplicial complex on $V(X_1)\dot{\cup}V(X_2)\dot{\cup} \ldots \dot{\cup} V(X_n)$, but for which selecting one point from each $X_i$ doesn't give a simplex? (e.g it could give a sphere)

Would such an object make sense? (e.g. be well-defined, not hopelessly horrible topologically)

I would also be glad to hear about similar generalizations in other contexts (as would be, for instance, generalizations to $r$-th secant varieties of affine varieties, provided the generalizations exist).

If such a generalization exists, what are typical situations where it is considered?

(I have been sloppy in several respects and I apologize for that. I hope the geometric idea of the generalization is more or less clear though.)