Tilda Swinton walks on water. The actress steps across silvery waves, ploughing the sea with an oar, dressed in a long green gown. Horses appear from nowhere, then ice floes, then seething volcanoes. A young man materialises to interpret her dreams, which might well include this bewildering film. Set in a post-apocalyptic Iceland and yet strangely medieval in tone, it is one of the oddest experiences you will ever have in a gallery.
In Volcano Saga (1985), the American artist Joan Jonas fuses myth with hyper-modernity – in this case a 13th-century Icelandic story envisaged using jump cut, green screen and video collage. This has been, very approximately, her signature style over the past half century. Primitive rites, strange ceremonies, the repeated gesture, dance or mantra: all are played out and enriched through the latest technology.
Jonas chalks a sun on a blackboard, rubs parts out to make a moon and then loops the metamorphosis on a television monitor. She appears inside a giant hoop, rolled along a beach like a kind of Vitruvian Woman, filmed on an early handheld camera. A massive central installation at Tate Modern features high-definition footage of a couple arguing over who owns what, photographs of the Sphinx, film of the Luxor hotel in Las Vegas, and a startling assembly of props, including what might be a minimalist sculpture of earth raked in geometric patterns. And all of this is a highly idiosyncratic retelling of Hilda Doolittle’s 1961 poem Helen in Egypt, following Helen’s perspective through the Trojan wars.
Born in 1936, Jonas trained as an art historian, studied kabuki in Japan in the early 60s, where she bought her first video camera, worked with sculptor Richard Serra and choreographer Trisha Brown. She is a veteran of New York’s downtown scene and a pioneer of performance art, with an unusual interest in new media. At Tate Modern, for instance, you can see formative 1972 piece Organic Honey, in which she appears variously dressed as a flapper, a belly dancer and in nothing at all, passing a mirror around herself, reflections from which are transmitted in close-up to a live audience. Then the audience itself is filmed. Watching, being watched, filmed, observed, reflected – it is a captivatingly strange cycle of viewing
But lord knows what is happening in Glass Puzzle II, where two women are filmed crawling about in tandem; or what is going on in Double Lunar Rabbits, where a Japanese man in a crude bunny outfit roams the streets. It is one thing to imagine the props in The Juniper Tree – a scarlet kimono, many scarlet paintings of severed heads – into a performance of the Grimms’ fairytale about a boy beheaded by his wicked stepmother; it is quite another to make sense of the Egyptian installation (or its contemporary political overtones, entirely lost here) without any of the live performers.

This is an occupational hazard, of course: how to revive the long-gone performance with anything more than its residue in objects and photographs. Jonas herself says that “the prop in my work is usually not a piece of sculpture in itself”, yet that is exactly how they are displayed at Tate Modern. It is perfectly possible to put on a Joan Jonas show that isn’t inert or mystifying. Her US pavilion at the 2015 Venice Biennale was a free-flowing fusion of films, objects and drawings that enchanted everyone who saw it.
One approach is to ignore the flannel panels that insist on historical context without edging anywhere close to the impact or meaning of each piece and simply wander free among the galleries, letting the work cast its peculiar spell. Dogs, shadow figures, the evolution of drawing, birds both alive and depicted, the gods and their eccentric behaviour, time and tide running through the roughest day – the threads gather into a web of mythology. Her art has something to say about animism and the supernatural nature of time, in which we are all suspended. It slows down the mind and makes one listen to the voices in her head.
Ultimately what’s missing, though, is Jonas herself. But some sense of her force of personality comes through in a series of wooden boxes into which you peer as if into a miniature cinema. Here she is dancing crazily before an intrigued dog, or confronting a version of her younger self or appearing before the camera, daily, to bid us good morning and later good night. The span and toll of 24 hours is compressed, over and again, in the life of this formidably purposeful woman. I couldn’t tear myself away.

Anthea Hamilton has done something completely unexpected at Tate Britain, not least because her previous outing there – the giant bare bum, Instagram moment of the 2016 Turner prize show – was so ostentatiously crude. The British artist, born in 1978, has transformed the stone canyon of the Duveen Galleries with a delicate originality into a kind of parallel universe. The neoclassical architecture is all still there, but inflected by contemporary white tiles that render it familiar and homely. Semi-abstract sculptures from the museum’s collection are now presented on structures more like beds, baths and even, at one end, a kind of tiled pool, instead of the usual pedestals.
And through this spacey new interior strange figures gradually move, furling and unfurling on table tops, drifting along the walls, reclining like the Henry Moore bronzes around them. One resembles a squash garlanded with pale spring blossoms; another, badger-striped, haunts the pool. They are supposedly based on a “lost” photograph of a person dressed as a vegetable, and sure enough, these exquisite couture costumes, made by Loewe, have overtones of eggplant, pumpkin and gourd. And so, of course, do the sculptures themselves.
The living beings animate the historic objects so that they appear newly outlandish, and equally organic. It is a gentle revelation. Through compelling choreography – performed on some days by Michael Clark’s dancers – shifted decor and beautifully absurd garb, Hamilton draws analogies between the still and moving figures in the art museum, bringing the spirit of modernism alive once more.

Star ratings (out of five)
Joan Jonas ★★★
Anthea Hamilton ★★★
• Joan Jonas is at Tate Modern, London, until 5 August; Anthea Hamilton at Tate Britain, London, until 7 October.