Shakespeare: Staging the World puts the Bard on the streets of his time. It illuminates the world he lived in, and the worlds he imagined, through material objects and images. But the British Museum show's not just for theatre fans. It's a glittering window on the Renaissance, full of fabulous artworks seen in a new way – through the rich glass of Shakespeare's poetry.

Wenceslas Hollar's brilliant 1647 panoramic map of London shows Shakespeare's Southwark, while a dazzling example of Venetian glass – spangled with white threads spun into its texture as it was fired on the island of Murano in the 1500s – takes us to his imagined Venice. Paintings by Titian and Catena expand on the allure of Venice for Shakespeare, who set two plays there. And a Venetian courtesan's shoes witness the city's reputation for sleaze.

A mega tapestry map shows Shakespeare's hometown Stratford upon Avon surrounded by the Forest of Arden that becomes the mythic location of As You Like It. There's a pair of antlers once worn by a stone-age shaman, just like the horns of Herne the Hunter in The Merry Wives of Windsor.

The British Museum's global perspective offers dynamic juxtapositions between Shakespeare's Britain and the world it was beginning to dream of colonising. It becomes striking that Shakespeare's last play was The Tempest, a fable of art and power set on a "primitive" island. Is Ariel – the spirit who serves the island's magical ruler Prospero – an image from the religion of Jamaica? In this exhibition, the airy spirit is actually personified by a 15th-century statue from this island. Meanwhile, Prospero's wizardry is illustrated by magical objects that belonged to the Elizabethan occultist John Dee.

It's great to see a portrait of John Donne in which the poet poses as an archetypal melancholy lover. The exhibition is good at showing the cultural architecture of the Renaissance world – the images that people lived by. It's tempting to say cliches, but these cliches were poetic enough to be used by Shakespeare, from the melancholy of love to the rage of jealousy.

Is Shakespeare the greatest British contribution to world culture, as this exhibition claims? It is hard to argue with that after seeing Mark Rylance as an awe-inspiring Richard III the other day at the Globe. Where the British Museum gives the historical low-down on Shakespeare, this astonishing performance by Rylance reveals the uncanny, dreamlike, apparently limitless power of his imagination.

Rylance reveals totally new dimensions to Shakespeare's royal villain. Flesh-crawling in his wan self-pity, grotesque in his sexual propositions, casual in his murders, this great incarnation of a monster makes you see that, however much Shakespeare wrote from a real timebound world, he transcended it to create demonic theatrical beings that live forever.