Realizing compatible pairs of transfer systems by combinatorial $N_{\mathrm{\infty }}$-operads

We investigate how the notions of pairings of operads of May and compatible pairs of indexing systems of Blumberg–Hill relate via the correspondence between indexing systems and $N_{\mathrm{\infty }}$-operads. We show that a pairing of operads induces a pairing on the associated indexing systems. Conversely, we show that in many cases, compatible pairs of indexing systems can be realized by a pairing of $N_{\mathrm{\infty }}$-operads.