In 1849 and 1856, the US House of Representatives elected a speaker by plurality vote after repeated failures to elect one by majority vote (62 failed ballots in 1849, 132 in 1855-1856). What is the procedure for switching from majority voting to plurality voting?
CBS News has published quite a good article on this.
December 1849, 31st Congress
On Dec. 19, the Whig caucus invited six Democrats and six Whigs to create a "Conference Committee," which met the next day, at which point 59 ballots had been cast with no speaker elected, according to "Fighting for the Speakership."
The "Conference Committee" decided there would be three more majority votes for speaker and if no speaker could receive a majority, it would go to a plurality vote. The rule change was brought to the House floor on Dec. 22, and it passed 113-105. The Democrats and the Whigs also decided to go back to supporting their original candidates, Cobb and Winthrop.
After no one received a majority in the next three votes, Cobb won on the 63rd ballot with a plurality, thanks to the newly enacted rule change.
February 1856, 34th Congress
On Feb. 1, 1856, Democrats decided to back pro-slavery Rep. William Aiken Jr. of South Carolina. Democratic leaders also said they would again support a proposal to elect a speaker via a plurality if nobody could win a majority in the next three ballots. On Feb. 2, the plurality resolution passed. Banks received the most votes in each of the first three ballots, but he fell short of a majority each time. Aiken then seemed poised to pick up the plurality on the fourth vote.
But Banks managed to secure 103 votes to Aiken's 100 because some of Aiken's expected votes fell through. Banks had won the speakership on the 133rd ballot, nearly two months after the first ballot had been cast.
It appears that the House can vote on a motion to allow plurality voting, and once it has done that, can elect a Speaker by that means.
I suspect that the current House would not have a majority for plurality voting at present. The Democratic candidate, Hakeem Jeffries, had a plurality on all of the first eleven votes, and none of the Republicans would be happy with him winning. McCarthy has had a plurality for the 12th and 13th votes, but the Democrats will want to make him work for the speakership.
Experience with Brexit has shown that a small group of hard-liners can make a party leadership's life very difficult for a long time.
Presumably they might switch to plurality if they can get promises from enough defectors to vote for McCarthy. But presently it seems these 20-some-odd detractors refuse to vote for McCarthy at all, regardless of concessions or situation. Otherwise it'd take a coalition of Repubs and Democrats to force it that way with a pick of their own. So, yeah, break out the popcorn, this doesn't seem likely to change soon. Jan 5 at 23:36
I have heard talk about it with the intended goal of forcing some of the hold outs to switch to McCarthy or let the Democrats pick the speaker. Though even if this does happen it would be short lived as they have the numbers to call for a vote to replace the speaker.– Joe WJan 5 at 23:36