On Thursday, Republicans began their third day supposedly in control of the House of Representatives but without a speaker – which means the House cannot function.

Over the past two days the Republican leader Kevin McCarthy of California lost six votes for the top job, because the extreme Maga right wing won’t support him.

To get their support, extremists in the House are demanding that any member be able to call a vote at any time to oust him, which would put McCarthy on a very short leash controlled by the Maga right (with Trump indirectly controlling them).

In effect, Trump and his loyalists would call many of the shots – on committee assignments, investigations (Joe Biden, Hunter Biden, the FBI), and key issues like raising the debt ceiling (they’ll demand that McCarthy refuse – imperiling the credit of the United States and catapulting the nation into an economic crisis).

Does this mean the rest of us have to sit back and allow a tiny minority of extreme rightwing Maga House Republicans controlled by Donald Trump to hijack congressional Republicans, who in turn will hijack the entire House, and thereby hijack much of Congress?

No. There’s an alternative, and House Democrats and the few remaining “moderate” Republicans should take it: come together to make someone like Michigan’s moderate Republican Fred Upton or Ohio’s David Joyce the speaker of the House.

There’s no rule that says the party in control of the House must decide on the speaker by themselves. All anyone needs to be speaker is 218 votes (or a majority of all members present), regardless of party.

House Democrats and moderate Republicans could come up with the 218 votes to put Upton or Joyce over the top.

Upton would be a good speaker but Joyce would probably be more acceptable to most current Republican representatives, even though the extremist Maga Republicans won’t have anything to do with him.

Joyce is the new chairman of the Republican Conference Group, a group you’ve probably never heard of (years ago it was called the “Tuesday Group”) because it flies under the radar. It’s a collection of the remaining 40 or so Republican moderates.

I say “moderate” only in comparison to the rest of the Republican House. The Conference Group at least wants the government to function.

Joyce is hardly a progressive. During Trump’s presidency, he voted in line with Trump’s stated position 91.8% of the time. And he voted against impeaching Trump for his role in the January 6 insurrection.

But Joyce is not a Maga Republican. He refused to sign the Texas amicus brief that tried to overturn the results of the presidential election. He was also one of the few Republican House members who did not object to the counting of electoral college votes on January 6, 2021.

Since Biden became president, Joyce has voted in line with Biden’s positions over 30% of the time.

He was one of 35 Republicans who joined all Democrats in approving legislation to establish the January 6 commission to investigate the storming of the US Capitol. He and 46 other Republicans voted for the Respect for Marriage Act, codifying the right to same-sex marriage in federal law.

Overall, Joyce’s politics are similar to the Democratic senator Joe Manchin’s. “Everybody’s a Joe Manchin,” Joyce said a few weeks ago.

Joyce wants to keep swing-district Republicans out of the harm’s way coming from the Freedom Caucus and other Maga conservatives.

He saw what happened to Ohio Republican candidates viewed as too close to Trump’s Maga wing: the state’s House delegation shrank from an eight-member edge for Republicans to just five because voters rejected several Maga candidates. “There’s some exotics that like chaos, they thrive in chaos because that’s how they get the media,” Joyce told the Washington Post.

Given that the likeliest alternative will be a Speaker Kevin McCarthy beholden to the extreme Magas, either Upton or Joyce should be elected speaker – and could be if only five House Republicans and all House Democrats support him.

What should Democrats ask for in return? A power-sharing agreement similar to the one agreed to in the last Senate, in which each party got the same number of seats on all committees.

The deal will enable the government to function and will simultaneously repudiate the Maga extremists. It’s a good deal for America.

• Robert Reich, a former US secretary of labor, is professor of public policy at the University of California, Berkeley, and the author of Saving Capitalism: For the Many, Not the Few and The Common Good. His new book, The System: Who Rigged It, How We Fix It, is out now. He is a Guardian US columnist. His newsletter is at robertreich.substack.com