We introduce a functorial construction $mathsf{C}$ which takes unitary magmas $mathcal{M}$ as input and produces operads. The obtained operads involve configurations of chords labeled by elements of $mathcal{M}$, called $mathcal{M}$-decorated cliques and generalizing usual configurations of chords. By considering combinatorial subfamilies of $mathcal{M}$-decorated cliques defined, for instance, by limiting the maximal number of crossing diagonals or the maximal degree of the vertices, we obtain suboperads and quotients of $mathsf{C} mathcal{M}$. This leads to a new hierarchy of operads containing, among others, operads on noncrossing configurations, Motzkin configurations, forests, dissections of polygons, and involutions. Besides, the construction $mathsf{C}$ leads to alternative definitions of the operads of simple and double multi-tildes, and of the gravity operad.
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