Reading Didion & Babitz is a bit like being held hostage. At the outset, I very much wanted what it appears to offer: an account of a friendship between two uncommonly fascinating American writers, Joan Didion and Eve Babitz, inspired by the discovery of a cache of letters found hidden at the back of the latter’s wardrobe after her death, aged 78, from Huntington’s disease in 2021. I guess you could say I was tied to my chair.

After a while, though, it came to me that these women had not, after all, engaged in much of a correspondence; the letter offered as bait at the start of Lili Anolik’s book, in which Babitz says mean things to Didion (“Could you write what you write if you weren’t so tiny, Joan?”), had, it turned out, never been posted. At which point, my disappointment was severe. I wanted to bust right out of the airless room in which I’d been kept for 190-odd pages, listening to Anolik’s annoying, digressive, smart-alecky prose – a style known to me as High 21st-Century Frantic American. Quickly, someone, open a window! Let me out of here.

One antidote to Frantic American is a long, cool dose of the very unfrantic Didion. But I had to keep going, if only to serve up here the few morsels of gossip Anolik’s text delivers – and, yes, they are mostly only gossip, even if I am with Babitz when it comes to the notion that whispers and tittle-tattle are just “some devious woman’s trick” (this is a mischaracterisation perpetrated by men, as she once wrote). The trouble is, perhaps, that while Anolik is a Babitz superfan (having managed to befriend the impossible Eve, a piece she wrote about her for Vanity Fair in 2014 boosted Babitz’s rediscovery; after it, New York Review Books Classics began republishing her autobiographical stories), she “naturally roots against” Didion, a woman she describes as “part princess, part wet blanket”.

Undercutting Didion’s ruthless stage management of her life has an interesting effect on her nonfiction

No wonder, then, that she takes some delight in airing rumours. Didion was an alcoholic, she suggests, while John Gregory Dunne, her husband of nearly 40 years, was probably gay. One of Anolik’s (secondhand) sources for the latter idea is Christopher Isherwood’s partner, Don Bachardy, who once told someone Dunne couldn’t stop staring at his crotch. The other is Bret Easton Ellis, who “had friends” who’d spotted Dunne “in certain gay bars at night, very drunk”.

This stuff is titillating, of course. By undercutting Didion’s ruthless stage management of every aspect of her life, it has an effect on her nonfiction in ways that may ultimately make it more interesting. The writer David Thomson tells Anolik about a call Didion made to her publisher, Sonny Mehta, the day after he received the manuscript of The Year of Magical Thinking, the memoir she wrote following Dunne’s death in 2003. “Will it be a bestseller,” she asked him, sounding “almost desperate” – a question that grazes like a sharp object. But the tinge of spite in the repetition of such anecdotes sits uneasily in a book that hopes to elevate Babitz. Dragging Didion down doesn’t in itself lift up her less famous friend.

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Were they friends, though? Their relationship strikes me as more casual than the word implies. When Didion and Dunne were living in LA after their marriage in 1964, Babitz, a goddaughter of Igor Stravinsky, was a designer of album covers and best known for having posed nude for a photograph with the artist Marcel Duchamp. She was part of the assorted arty, literary gang that used to visit the couple (like supplicants), first at their rundown mansion in Hollywood, and then at their place in Malibu, where she would marvel at Didion’s abilities as a hostess (she had Spode china).

The two were opposites: Babitz, an ex of the Doors frontman, Jim Morrison, was a sexually prolific seductress, while Didion had slept with only one man before her husband. Where the former was voluptuous (“boobalicious”) and direct, the other was slight to the point of emaciated, and given to calculated silences. Anolik sees them as two sides of a coin; her thesis is that Babitz’s Slow Days, Fast Company (1977) and Didion’s The White Album (1979) are companion pieces, utterly necessary to each other (both are essay collections that stake out Los Angeles/California). But whether this is true, let alone enough to form the spine of this raggedy double biography, is moot.

Didion certainly helped Babitz to get going as a writer, recommending her piece The Sheik to Rolling Stone. But their relationship broke down during the writing of Babitz’s first book, Eve’s Hollywood (1974), which Didion was editing (“I fired Joan,” said Babitz, with some grandiosity). Things cooled. Didion, the reader understands, did not have close friends. She had Dunne, with whom she forged both a marital myth and a literary brand. When Babitz burned half her body in a dreadful accident in 1997, the Dunnes sent a note whose gelid heartiness fairly takes the breath away; in truth, Didion no longer thought of Babitz at all. Their last encounter occurred in 2005 when Didion was appearing on an interview show and Babitz naughtily rang in.

Thereafter, their lives diverged with a starkness straight out of a Ryan Murphy show. After the loss of Dunne and her adopted daughter Quintana, Didion became a secular saint: a woman so famous, her death in 2021 was marked by such celebrities as Reese Witherspoon, Lynda Carter, Johnny Marr and Harry Styles (“you might almost be fooled into thinking the world cared about books or the people who made them”, as Anolik puts it). Babitz, meanwhile, retreated into obscurity, physical pain, domestic squalor and rightwing politics. Personally, I’m in thrall to them both: the one all heat even as the other glaciates. But yoked together like this, they’re each diminished. Neither of them would have liked this book. As I read, I sensed ghostly shudders: phantasmic rebukes from the great memoir workshop in the sky.

Didion & Babitz by Lili Anolik is published by Atlantic (£20). To support the Guardian and Observer order your copy at guardianbookshop.com. Delivery charges may apply