British Museum brings the Bard back to life with blockbusting Shakespearean objects show

Although it leads you past many strange and wondrous objects, from a bear's skull to a Jesuit's eye and a unicorn's horn, the British Museum's Shakespeare show begins and ends with a book. As you enter the old Reading Room, to the recorded sounds of London streets, the playwright's engraving from a First Folio of 1623 greets you with its enigmatic stare – and not just any First Folio, but a particularly fine copy that belongs to the Jesuits of Stonyhurst school.

It's a nice emblematic touch, given that the exhibition demonstrates both how the Stratford prodigy gave dramatic form to the divisions of his age – the murderous rifts of Protestants and Catholics prominent among them – and later served as an adhesive force that bound disparate people and cultures together. Hence the final item, the "Robben Island Bible": a copy of the Complete Works smuggled into the South African jail by a political prisoner under apartheid, Sonny Venkatrathnam. Inmates would debate the plays with passion and add a signature to passages that meant the most to them. The "Bible" lies open at the lines from Julius Caesar in which Caesar says, "Cowards die many times before their deaths/ The valiant never taste of death but once." Nelson Mandela signed them on 16 December 1977.

Laden but not overstuffed with riches – ideas as well as material treasures – Shakespeare: Staging the World aims not just to match artefacts with plays, nor to recreate the stage of Elizabethan and Jacobean times. It seeks to offer a visual and auditory route through the world the playwright channelled into his work. "The objects are not illustrations," curator Dora Thornton insists. "What we're trying to do is provide an intellectual context for them."

The globalised dimensions of this world – from a Sri Lankan jewel bracelet to a Moroccan envoy's portrait – signify not mere fancy imports but embedded parts of London, and British, culture in all its fast-evolving hybridity. Around 1600, a local craftsman pimped a lovely Iznik pot from Turkey into a handled jug. Thornton notes "how these exotic trade items have been customised and made even more outlandish".

Drawn from the BM's whole collection, and enhanced by loans, the show covers all the documentary and archaeological bases: maps, portraits, books, manuscripts, genealogies, even that poor bear's skull dug up from the site of the Globe, with claw-marks from the baiting dogs visible. Yet its soul lies in the space between the objects we inspect and words spoken by the ghostly video simulacra of RSC actors. In a section devoted to Rome and the Classical world, Paterson Joseph not only delivers a speech from Julius Caesar but holds a rare gold coin (on show) minted c.43BC to commemorate Brutus's conspiracy. In the dark-hued corners and corridors that evoke kingship and witchcraft at the time of James 1's accession, the "weird sisters" from Macbeth intone spells next to clan talismans from Scotland.

Not a comprehensive guide to Shakespeare's settings, subjects and themes, the exhibition brings to multi-dimensional life "the issues that fired his imagination, and the questions that were problematic for his audiences". So, for The Merchant of Venice, we view not merely the city's luxury exports in metal, jewellery or glass but a Sabbath lamp from Windsor, dated back to before the expulsion of Jews from England in 1290. With Othello, the meaning of the "Moor" becomes manifest not only through images of Africans that may (Thornton suggests) mark "the beginning of a black European identity". It resides in the ambiguity of the Adoration portrait of a "turban'd Turk" by Catena, prostrate before the Madonna and infant Christ. It doesn't seem to show a conversion but must he, like Othello, endlessly affirm his faith? Like the words that frame them, these artefacts keep their secrets.

Mandela's message comes at the exit to the light-filled final space. Here The Tempest and its brave new world of wonders opens our eyes to the vistas of discovery – and colonisation – that lay just over the London horizon: the globe beyond the Globe. Prospero and Caliban (Ian McKellen and Amer Hlehel) present rival perspectives of this age of marvels, in keeping with the show's mood of kaleidoscopic ambivalence. "Each person will have a slightly different set of kaleidoscopes," says Thornton. If reverent Bardolatry has no place here, neither does reductive historicism. The mystery of this imagined realm persists, even as Prospero's enchantment breaks on the rocks of Robben Island and (as the curator comments) "you leave having been taken into the modern world". As for that "unicorn's horn" – seven feet long – it came, as they all did, from a narwhal: an Arctic whale.

Shakespeare: Staging the World, British Museum, London NW1 (020 7323 8181; britishmuseum.org) to 25 November