British actor Tom Burke has been spending a lot of time with Cate Blanchett lately.

Burke (“Furiosa: A Mad Max Saga”) is currently starring alongside the Oscar winner in a new adaptation of Anton Chekhov’s classic “The Seagull” at London’s Barbican, and the two also worked together on Steven Soderbergh’s upcoming espionage thriller “Black Bag.”

In “The Seagull,” Blanchett stars as Arkadina, a celebrated actress whose larger-than-life presence dominates both the stage and her personal relationships. Arriving at her family’s country estate for the weekend, she finds herself caught in a storm of conflicting desires. Her playwright son, Konstantin (Kodi Smit-McPhee), struggles to step out of her shadow as he pursues his own artistic ambitions and her lover Trigorin (Burke) becomes the object of affection for the aspiring young actress Nina (Emma Corrin).

Duncan Macmillan and Thomas Ostermeier’s adaptation caused the cast to collectively embrace...