The immersive theater company’s production invites up to 600 spectators to roam freely around a loose re-creation of the siege of Troy’s aftermath.LONDON — It’s unusual in a live performance to construct your own narrative, shaping the event as you see fit. But that has long been part of the appeal of Punchdrunk, the ambitious immersive theater company whose latest show, “The Burnt City,” opened here last week.There are no assigned seats, or even spoken words, in the company’s first London project in nine years. Instead, the co-directors, Felix Barrett and Maxine Doyle, encourage up to 600 spectators to roam two onetime munitions factories (and a new structure conjoining them) and make of the occasion what they will. In my case, that meant being enthralled more often than I was baffled; others may well have the opposite response.Taking as its topic the fall of the ancient city of Troy, the show includes in its cast of characters Agamemnon, Clytemnestra and Cassandra, and it dramatizes the cycle of vengeance that follows Paris’s abduction of the famed beauty Helen.The characters, played by a hard-working company of 28 who perform their scenes in a loop, aren’t identified, so you’re left to work out who might be the Trojan queen, Hecuba, or her ill-fated son Polydorus, whose murder is one of several in a narrative full of grief. If you happen recently to have read “The Iliad” or the tragedies by Euripides and Aeschylus that underpin this venture, so much the better.Wearing masquerade masks, as is the Punchdrunk norm, we begin in a hall of display cases filled with artifacts from a 19th-century excavation of supposed Trojan ruins by the German archaeologist Heinrich Schliemann — pottery, libation bowls, headgear and other items. They form what Barrett has described as a “decompression zone” to help us shake off the outside world and plunge us into a bygone civilization. (To that end, cellphones are placed in sealed bags during the performance.)Leaving the dimly lit gallery, we embark on our chosen journey: Turn one way for Troy, the other for Mycenae, the Greek military stronghold that vanquished the smaller city around 1250 B.C.“The Burnt City” is the first London project in nine years from Punchdrunk, which also created “Sleep No More.”Julian AbramsMost of the action plays out in the capacious, high-ceilinged rooms of the warehouse representing Mycenae, including Agamemnon’s sacrifice of his daughter Iphigenia and his eventual murder — here graphically depicted in a shower. (Nudity is presumably one of several reasons that playgoers ages 16 and 17 are allowed entry only with “a responsible guardian.”) Stephen Dobbie’s mood-setting sound design thrums ominously throughout, and at several points we encounter some frenzied, furious dancing in which Doyle, a noted choreographer, lets her performers cut loose.Troy, by contrast, is a deliberate mash-up of eras and references, and the exemplary design team of Barrett, Livi Vaughan and Beatrice Minns get to have some macabre fun. This neon-lit labyrinth features a department store called Alighieri’s that evokes the underworld of Dante, and piles of bones to remind us of the siege of Troy’s carnage. An illuminated sign advertises “finest fake flowers,” for anyone who might want to pay respects.In contrast to previous Punchdrunk shows — like the company’s signature New York success, “Sleep No More” — there is little buttonholing of individual playgoers for one-on-one encounters (perhaps not so desirable in the age of social distancing), and the proceedings don’t build to the usual galvanic finale. You depart impressed by a concerted appeal to the imagination, though maybe another go-round is needed to fill in the gaps.Punchdrunk asks audiences to expect the unexpected, and so, in its way, does “Daddy: A Melodrama,” the Jeremy O. Harris play running through Saturday in its London premiere at the Almeida Theater. Directed, as in New York in 2019, by Danya Taymor, the production places an infinity swimming pool downstage — not the first thing you expect to see upon entering an auditorium.Terique Jarrett and Sharlene Whyte in Jeremy O. Harris’s “Daddy: A Melodrama,” directed by Danya Taymor at the Almeida Theater.Marc BrennerSpectators in the first few rows shield themselves as the actors splash about, with frontal nudity, as in “The Burnt City,” presented unselfconsciously. The frolics serve a story that grips across nearly three hours, even as it tilts after the intermission toward the melodrama of the title. Telling of a Black American male artist and the older white “daddy” who acts as the younger man’s patron and lover, Harris’s play is a parable of possession, in which people can be owned, just as art can.The charismatic Danish actor Claes Bang (now onscreen in “The Northman”) plays Andre, a European art collector based in Los Angeles, and the hugely gifted Terique Jarrett, handed the driving part, plays Franklin, the mid-20s boytoy who makes dolls of varying sizes — and who may represent a doll of sorts to Andre.Complications arise when Franklin’s deeply religious mother, Zora (Sharlene Whyte, commendably fierce), arrives for a visit only to voice displeasure with the lifestyle her boy has chosen. “What happened?” she demands to know of the Bible-quoting son who once sat on her lap in church. Franklin’s chums take their own poolside view of events: “So I guess since Mom’s a no-go,” says Max (the musical theater actor John McCrea, in waspish form), “Daddy has to suffice.”Whyte’s Zora faces down her son with an outsize grandeur worthy of Punchdrunk at its most heightened. The male leads, meanwhile, expertly chart the changing dynamics of a liaison at risk of burning itself out. Franklin, for all the fuss made over him, looks poignantly set on a path toward loneliness, left with not so much a burnt city as a scorched soul.The Burnt City. Directed by Felix Barrett and Maxine Doyle. One Cartridge Place, through Dec. 4.Daddy: A Melodrama. Directed by Danya Taymor. Almeida Theater, through April 30. More