Is it okay to elope with another man’s wife, if that other man turns out to be your father? Would you forgive a lover who only stopped his unfaithful ways after you convinced him you were a ghost? Could you trust a fortune-teller to lead you through your life?

It’s probably best not to think too hard about the little moral and logical pickles our characters confront throughout L’Ormindo as the opera presents a tangle of unconvincing associates and lovers. The unlikely tale begins as Princes Ormindo (Samuel Boden) and Amidas (Ed Lyon) travel across northern Africa to offer their support to the elderly King Ariadenus (Graeme Broadbent). When matching photos of an ‘unmatched beauty’ are revealed by the young men, our princes position themselves as love rivals – and our King discovers that the affections of his spirited young wife, Queen Erisbe (Susanna Hurrell), are somewhat scattered.

Fortunately, this sharp and imaginative collaboration between Royal Opera House and Shakespeare’s Globe leaves only its characters in a muddle. Using a comically flawed narrative and a rich, dazzling space to good advantage, this opera trills on the right side of the thin line between authenticity and kitsch. While the quality of the opera is clear to see, the original, seventeenth-century plans for this indoor playhouse leave little guidance on where to position surtitle boards – making drama and storytelling the focus. Responding to this alternative venue, the Royal Opera’s Director of Opera, Kasper Holten, proceeds “treating the opera as text set to music”. Throughout, the performers’ musical brilliance remains understated.

Relentlessly ornate, the new Wanamaker Playhouse offers a distinct challenge to designers: how do you place a stamp of individuality on your production, when a stunning landmark of a venue threatens to steal the show? Anja Vang Kragh’s costume designs provide the answer, with an edgy sultriness that riffs on the macabre glamour of this opera. The designer interacts well with Christopher Cowell’s neat translation of the Giovanni Faustini’s libretto, complementing a toxic world where beauty has an obvious ugly streak. A definite, gendered cynicism bleeds through the piece’s attitude to contemporary cosmetics: “Each time you kiss a woman you swallow poison.”

Queen Erisbe gets the best costume – a vibrant, elegant gown that sweeps up at the back to double as a bedsheet, cleverly ornamented by two pillows and a headboard. The queen’s whimsical dress serves as costume and setting; what it doesn’t do is cover the back of her pink French knickers, underlying L’Ormindo’s naughty flair. Later, soprano Joélle Harvey takes to the stage as Lady Luck, in Lady Gaga’s metallic haute couture dress and Lady Liberty’s overblown crown. Paradoxically, by channelling cartoonish influences onto the opera’s superficial caricatures, Vang Kragh brings depth and richness to the show.

As beeswax candles illuminate a colourful stage that Shakespeare’s Globe insists is “an archetype, rather than a replica”, a production here could be so tacky. If done frivolously, this could certainly feel like Disneyland Baroque. Thankfully, the collaborators have done well to highlight the grotesque and playful in this work, bringing a heavily-stylised storybook quality to their adaptation. As the musicians tune up, our players relax onstage in dressing-gowns, whispering and gossiping with each other and applying their make-up. When the music begins, the actors look surprised and pack up, engineering a sharp transition into what is a fictional world that simply drips with allegory.

L’Ormindo is playing at the Wanamaker Playhouse until 12 April. For more information and tickets, see the Shakespeare’s Globe website.