Rep. Mike Rogers (R-Ala.), who chairs the Armed Services Committee, told reporters Thursday after a closed-door House GOP meeting that he wants Minority Leader Hakeem Jeffries (D-N.Y.) to tell Republicans what concessions they’ll have to make for Democrats to help them elect a speaker—underscoring the chaotic race to find a new House leader as Republican options grow short to overcome an intraparty impasse.
It is kind of is possible, because the Speaker can only conduct business in the House if a majority agrees. There are procedural votes all the time, for things like the agenda for the session, which need this majority.
After the Debt Ceiling vote, the far right had a temper tantrum and stopped voting with McCarthy for a week or so, and nothing could get done until Kevin gave them back their binkie so they would start voting the right way again
Presumably, in any agreement to share power, whatever coalition voted for the Speaker would be expected to keep voting in his favor on those procedural votes, otherwise the whole thing stops. Which also means that the Speaker has some incentive to keep his coalition in his camp. In a typical party-based majority this should be easy, but Matty and Jimmy get mad when you take away their binkies.