HOLLOW CROWN: Tim Pigott-Smith plays the Prince of Wales elevated to monarch in “King Charles III.” (Joan Marcus)

NEW YORK–And you thought the Plantagenets and Tudors had the royal market cornered on theatrics.

Wait until you meet the battling, plotting, fulminating, equivocating Windsors of “King Charles III,” Mike Bartlett’s sublime peek past the reign of Elizabeth II and into the turbulent monarchy her son inherits. Fortified by a sensational central performance by Tim Pigott-Smith as Charles, Bartlett scores one remarkable dramatic coup after another, in an evening so convincingly modeled on Shakespeare that never for a moment does an audience question his choice of having Charles and William and Harry and Kate and Diana converse in iambic pentameter.

That’s right, even the late Princess of Wales materializes here, floating in from the great beyond like some chic, furtive version of Banquo or the Ghost of Hamlet’s Father, to unsettle the denizens of Buckingham, Kensington and Windsor palaces. For sure, this evening could easily have been a facile or worshipful gimmick, another cheap excuse to gawk at the royal family. But be assured that “King Charles III,” directed with panache by Rupert Goold, has a serious case to make, about the potential moral and legal relevance of the sovereign in 21st Century Britain. And just as incisively, the London-born play, which had its official Broadway opening Sunday night at the Music Box Theatre, wants us to consider the Windsors, particularly the younger generation, not merely as aristocratic anachronisms, but as stealthy, worldly power players, mindful of their institution’s historic role and determined to preserve it.

It’s unlikely you’ll ever again imagine, for instance, that there might not be much going on behind the charming porcelain features of Prince William’s wife, Kate, the Duchess of Cambridge. As portrayed by Lydia Wilson, she’s a woman with an undercoat of steel, one fully aware of the benefits of casting an alluring public spell, and invested in her husband’s regal trajectory with a fervor bordering on Lady Macbeth’s. Similarly, the princes, William (Oliver Chris) and Harry (Richard Goulding) emerge here as far more interesting people than they’re given credit for in lazier, “fact-based” treatments of their lives.

Prince Harry’s penchant for pub crawls, fast food joints–and anonymity–stamps him as a kindred spirit with Shakespeare’s own Prince Hal, who roams the inns of Cheapside in “Henry IV Part 1” to slake his thirst for the romantic roughness of life among the working classes. Chris’s William, meanwhile, gets to reveal his own princely mettle in a thrilling, no-holds-barred encounter with Charles, during which he delivers a passionate, audience-pleasing denunciation of his father’s shabby treatment of Diana, his mother.

Bartlett, known best in these parts as the author of “Cock,” a play about the psychological battle for the heart of a man of vacillating sexual preferences, counts this time on an audience that is invested heavily in the classics or the royals. Exploiting these affections is of course always a safer bet in London, where “King Charles III” was in fact a critical and popular hit. We Yanks do have a soft spot for stories about the travails of Elizabeth and her brood, a fascination Helen Mirren has successfully mined, in award-winning impressions of Elizabeth II on both screen (“The Queen”) and stage (“The Audience”).

Time will have to tell whether “King Charles III,” despite its high entertainment value, is quite as natural a fit on Broadway. Bartlett takes a far less sentimental approach than did screenwriter/playwright Peter Morgan in the Mirren vehicles; he demands of an audience a more sophisticated understanding of how a constitutional monarchy functions, and expects us to have some intellectual curiosity about this eccentric man, nearing 70, who’s been waiting his entire life to take the throne.

Staged on set designer Tom Scutt’s spare, darkly handsome platform, ringed by a mural of ancient, painted faces worn down by time, “King Charles III” is indeed informed by the world’s fading appreciation of royal prerogative. It focuses on a crisis that erupts immediately after Elizabeth’s death, when the obstinately principled Charles refuses to sign a piece of legislation passed by Parliament that places onerous restrictions on the press. (The irony is not lost here, of a man hounded and pilloried by the news media becoming its defender.)  This single act of defiance, though, lights a fire that Charles cannot extinguish. Having had the king throw the question of his power in their faces, the prime minster (Adam James) and Parliament set out to squash him and the monarchy once and for all. The “House of Cards”-style battle is downright fun to watch.

LONG LIVE THE KING? Tim Pigott-Smith and the cast of “King Charles III.” (Joan Marcus)

Though the actors portraying the other well-known royals bear strong physical resemblances to their real-life counterparts, Pigott-Smith looks nothing like Charles. And yet, his is the most captivating embodiment of all. That’s because in his perplexed and troubled countenance he makes the tragic dimension of Charles’s imagined reign feel so compellingly inevitable. Given the generous spirit this Charles shows, especially toward the sons he professes to want close by, Pigott-Smith’s character does seem a more emotionally available variation on the real man. This does help in conveying the poignance of Charles’s predicament: succeeding one of the most durable and beloved leaders in British history. “I never thought I’d see her pass away,” a character remarks mournfully. “I felt the same,” replies Charles. And you can’t help but feel for a man compelled to wait a lifetime for a vacant seat.

“King Charles III” raises many intriguing issues, one of them being about the value of an unelected figure, serving as a moral counterweight to his government. Another is about Charles himself, and what kind of stuff he’s really made of. As in many other matters, the play suggests finding some answers in the work of the maestro of the history play–in this instance, in his “Richard II.”

In concert with composer Jocelyn Pook, lighting designer Jon Clark and Scutt, who also created the costumes, Goold does a marvelous job conducting the proceedings, especially in the solemn customs and ceremonies that frame the evening. Wilson, Goulding and Chris contribute such admirable work that their real-life palace counterparts might even be flattered. Other supporting performances are exceptional, too, from Tafline Steen’s turn as an anti-monarchist art student Harry falls for, to Miles Richardson’s portrayal of a royal handler whose survival skills are better-honed than his king’s. For all involved, the production is a crowning achievement.

THE NEXT GENERATION: Prince William (Oliver Chris) and Kate (Lydia Wilson) in “King Charles III.”(Joan Marcus)

King Charles III

By Mike Bartlett, directed by Rupert Goold. Sets and costumes, Tom Scutt; music, Jocelyn Pook; lighting, Jon Clark; sound, Paul Arditti. With Sally Scott, Margot Leicester, Miles Richardson, Anthony Calf. About 2 hours 35 minutes. Tickets, $37-$227. At Music Box Theatre, 239 W. 45th St., New York. Visit telecharge.com or call 212-239-6200.