Is Hakeem Jeffries the real Speaker of the House of Representatives?

No, he's not. Jeffries remains the Democratic House Minority Leader. 

Republican Mike Johnson is the Speaker, leading a one vote majority with Marjorie Taylor Greene and other far right lunatics nipping at this heels. It's a precarious perch. But Jeffries has managed to win a commanding position in the current Congress by means of cross party legislative legerdemain.

Russell Berman describes the arrangement with Jeffries that is keeping Johnson in his current job:

... in an unusual statement, the leaders of the Democratic opposition emerged from a party meeting to declare that they would rescue Johnson if the speaker’s main Republican enemy at the moment, Representative Marjorie Taylor Greene, forced a vote to oust him. ... by thwarting Greene’s motion to vacate, Democrats hope they can ensure that Johnson will keep turning to them for the next seven months of his term rather than seek votes from conservative hard-liners who will push legislation ever further to the right.

Rep. Hakeem Jeffries, D-NY

For the sake of historical memory, it seems worth recalling that former California Assembly Speaker Willie Brown pulled off similar legislative wizardry in the mid-1990s. Brown served as Democratic Assembly leader for fifteen years, using his near absolute power over his caucus to advance civil rights and to raise the cash to elect Democrats. The push for legislative term limits in the state -- passed by voter initiative in 1990 -- was often described as the only way to pry Willie Brown out of the job.

In 1994, as the new term limits led to a bit of legislative turnover, for the first time in decades, Republicans seized a one vote majority in the state Assembly. But Brown wasn't about to let them take over "his" house. First he persuaded a dissident Republican, Paul Horcher, to turn independent and vote to keep him as Speaker. Horcher was quickly recalled by his Republican district. (Horcher stayed in Brown's orbit; when Brown was later mayor of San Francisco, Horcher worked in department of parking and traffic.)

After Horcher was deposed, Brown worked his magic to hang on yet longer. He persuaded another disgruntled Republican member, Doris Allen, to take the title of Speaker, for the next six months. But her power consisted of Brown's control of the Democratic Caucus. After this, Dems again won the Assembly majority and that was that.

The California legislature only saw the last of Willie Brown when term limits sent him off to be "da Mayor" in San Francisco.

If Democrats win the House in November -- a strong possibility if we do the work -- then Hakeem Jeffries will be elected the actual Speaker. But he's certainly proving a worthy successor to former Speaker Nancy Pelosi and her mentor Willie Brown in this interregnum. 

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I have to add that Willie Brown's later influence was extremely malign on the city I love . His mayoral tenure was both corrupt and indifferent to majority who were too poor to offer much to him and his friends. He governed for the civic fat cats and left of a legacy of subsequent lesser imitators. But he sure was a phenomenal leader of the Assembly in his day.