From the east, the elevation presents a slim, unadorned, milk-white concrete block, nine stories high, punctured by loggias — a signpost, like the traditional village bell tower, rising above a low, scruffy neighborhood.

To the north, where the facade meets Milan’s skyline and becomes mostly glass, cantilevering over the street, the block breaks into a zigzag of shifting floor plates, rectangles and trapezoids, the whole building wedged onto a triangular plot.

The south end makes plain how the structure stands up. An ensemble of enormous cables encased inside a giant beam counteracts the thrust of all those heavy, cantilevered concrete decks. Like a sword in a stone, the beam angles from the top of the tower through the red-tiled roof of an adjacent former warehouse, anchoring in the floor below.

In the Arthurian legend, the wizard Merlin put the sword in the stone. Mr. Koolhaas must be Merlin, I suppose. That makes Miuccia Prada, the Lady of the Lake.

The tower completes the arts campus OMA has spent the past decade conceiving for the Prada Foundation. An offshoot of the global fashion conglomerate, dedicated to contemporary art and culture, the foundation commissions new art, presents exhibitions and organizes film festivals and other events. It also oversees the vast art collection Ms. Prada and her husband, Patrizio Bertelli, have put together. For years, it operated in far-flung locales.

In 2008, Mr. Koolhaas and a partner, Christopher van Duijn, were enlisted to reimagine a former, turn-of-the-century distillery Prada owned as the foundation’s permanent home. Walled-in, abutting a weedy stretch of railroad tracks, the distillery was a picturesque assortment of dilapidated stables, a bottling facility, a carriage house, some offices and warehouses.

The architects cleared away some of the old buildings, refurbished others. They built new ones. The tower was the last piece of the puzzle.

Without it, the site first opened to the public in 2015. It featured about 120,000 square feet of new or reconfigured exhibition space; a new cinema; a new two-story Miesian pavilion of wide open gallery spaces, called the Podium, the whole building clad in light, shimmery panels of foamed aluminum, an automotive and medical industry material also used for bomb blast absorption that looks a little like rough stone. There was even a 1950s-style Italian cafe straight out of a Wes Anderson movie.

That was because Wes Anderson designed it.

Chameleons themselves, Mr. Koolhaas and Ms. Prada made natural confederates. She was the famous communist turned high-fashion mogul whose empire evolved from bags and backpacks constructed out of an industrial nylon lining material. He was a prophet of global cities who declared the countryside his real passion after everyone else jumped on the urbanist bandwagon.

Her clothes always seemed less about what men desired than what whet her creative appetite. He was once invited to propose an expansion for the Museum of Modern Art and thumbed his nose at the selection committee by suggesting a billboard that said “MoMA Inc.” They were both contrarians and closet optimists.

And they shared a sense of humor. At one time it was rumored that Prada might back the Dutch architect for a seat in the Italian Parliament.

The foundation became their love child. It is unlike the eye-popping art gallery Frank Gehry designed for the Louis Vuitton Foundation beside the Bois de Boulogne in Paris, with its billowing glass sails, conjuring up flounces of silk and memories of Bilbao. The Prada campus feels, by comparison, world-weary, sneakily luxurious and — especially with its new tower — a mini-city, fragmentary, full of craft and secrets. Cities enshrine history and agitate for change. They’re forever unresolved.

This has been Mr. Koolhaas’s mantra. It is reflected in a foundation that’s neither a preservation project nor a tear-down-and-build-new venture. Its mode is bricolage. More is more. Both is better.

Cities are theaters and shape-shifters, too. I’m vaguely reminded of the old Cinecittà studios outside Rome, where Fellini worked and Mr. Anderson has made films. A stable house in the former distillery now resembles the cabinet rooms in old master museums. A tiny, Alice in Wonderland door opens onto an immense warehouse, 60 feet high and 200 feet long.

And a building nicknamed the Haunted House is slathered in gold leaf, like an early Renaissance panel painting. (“A very cheap cladding material,” Mr. Koolhaas has insisted, “compared to marble or even paint.”)

“There is no difference between gold and rags,” Michelangelo Pistoletto, the veteran artist, once said. Mr. Pistoletto made his bones in the 1960s as a founder of Arte Povera, the Italian twist on post-Minimalism. Writing in 2001, after Mr. Koolhaas’s Prada shop opened in downtown Manhattan, the critic Herbert Muschamp noted Prada’s philosophical roots in Arte Povera.

Mr. Muschamp recalled how Art Povera consisted of “old bedding and tar-stained rope” displayed “in barren, out-of-the-way locations.”

Somehow, he added, “you always needed a private jet to get there.”

Up to a point, that describes the foundation, with its fetishized lowdown materials like chipboard and orange construction fencing and slightly out-of-the-way location, south of the city’s center.

Arte Povera isn’t the only ghost of midcentury modernism inhabiting the project — there’s the twee cafe, with its Formica furniture and veneered wood paneling; the new tower restaurant, with its furniture bought at auction from New York’s Four Seasons; the cinema, with chairs imported from ’70s-era Milanese movie houses; and the sun-baked, deeply shadowed squares, conjuring up de Chirico.

At the same time, there are the custom sheets of very modern translucent polycarbonate and aluminum handrails milled like Ferraris. There are the oak wood box-on-end pavers and the repurposed metal prison grates painted lime green, which serve as screens in the coat checks and bathroom stalls.

Some visitors have complained the layout doesn’t tell you where to go. You find your way around it. Like in a city. I think that is a virtue.

But until now the project was missing its cornerstone where the 200-foot-high, 22,000-square-foot tower, or Torre, was meant to rise. Delays in construction stretched three years. They ended up allowing time to refine the design.

The tower’s six, stacked gallery floors were created as full-time showcases for Prada’s private art collection. They’re reached through a small, open-air lobby like a disco ball, with flashing screens and a dizzying cutout in the ceiling to reveal the building core’s scissored stairs. One flight up, mirrored bathrooms, industrial sinks and a patterned floor summon to mind Pierre Chareau and Superstudio.

The galleries above are one to a floor, no two rooms alike, each taller than the last, their layouts shifting with the floor plates, the lowest gallery, 9 feet high; the topmost, 26 feet high.

The middle-floor galleries end up feeling the nicest, proportion-wise. But the whole building is one narrative. As Federico Pompignoli, OMA’s project manager, has said, the tower is “an attempt at the white cube defying its own boringness.”

Much credit here goes to him. He oversaw every inch of construction and it shows. Elevator cabs clad in backlit slabs of rose and green onyx suggest medieval reliquaries. I am told blacksmiths from a tiny shop outside Milan hand-tooled the restaurant’s exquisite bar, sliding doors and custom-embossed the anodized aluminum panels on the terraces that look like expensive Lego pieces.

I kept running my hands over the tower’s concrete walls. Infused with Carrara marble and poured by construction workers who wore white gloves, they feel smooth as silk.

The big rectangular and wedge-shaped galleries, windows alternating between panoramas of the city and narrow views over the campus, accommodate best the large-scale works in the inaugural show, “Atlas.” It features Jeff Koons, Mona Hatoum, Michael Heizer and others. Check out the restaurant if you go. Works by Carsten H?ller and Lucio Fontana are on permanent display.

Then I wandered into Mr. Anderson’s cafe and ordered what may be the most delicious sandwich I have eaten in my entire life.

Private museums are mostly vanity projects. Few invent social spaces. It may be the ultimate tribute to Mr. Koolhaas and OMA to say that the Prada campus works. The plazas are poetic. The galleries are practical and varied.

Ms. Prada should be pleased and maybe a little worried. It’s up to the foundation to program these spaces for generations to come.

Architecturally speaking, there’s a lot to live up to.