2018–2024: International recognition
Kulig's international breakthrough came in 2018 when she starred in Pawlikowski's historical drama film Cold War. The film marked their third collaboration, which prompted Pawlikowski to call her his "muse". The part of Zula, a woman in post-war Poland who joins a folk music touring group, was once again written by Pawlikowski with her in mind. Kulig was drawn to the role as it required her to sing and dance, allowing her to showcase her talents; she trained with a folk group for half a year in preparation for the part. In the film, she notably performs the traditional Polish folk song "Dwa serduszka" ("Two Hearts"), termed by Vulture as "the movie's unofficial theme song". Mark Kermode of The Observer wrote that she "delivers a star-making performance of astonishing range and depth", and Mick LaSalle of San Francisco Chronicle stated that "she takes the role of a lifetime between her teeth, chomps on it, pounds it into the ground and never lets go for a second." According to Scott Feinberg of The Hollywood Reporter, "in a fair world ... Kulig would not only be a serious contender for a best actress nom, but would quickly become an in-demand actress in Hollywood, too." Kulig received the Best Actress Award at the 31st European Film Awards, and the Polish Academy Award for Best Actress for her performance. That same year, she played local priest's pregnant girlfriend in Wojciech Smarzowski's film Clergy, which went on to become the highest-grossing Polish film of all time. Due to its themes of child abuse, corruption and alcoholism in the Catholic Church in Poland, the film caused controversy especially among right-wing groups in the largely Catholic country and led to a nationwide debate about the role of the church in the Polish society. In 2019, she was cast in the recurring role of Johanna, titular character's mother, in Amazon Studios series Hanna.
Kulig starred as Polish jazz singer Maja in Damien Chazelle's 2020 Netflix musical drama series The Eddy. The character was originally intended to be American, but after meeting with Chazelle the part was rewritten for her. In 2021, Kulig was cast as the lead in TVN thriller series Spider's Web playing a spider behaviour specialist trying to unravel the mystery of the death of her mother and sister. She served as one of the jurors of the Un Certain Regard section at the 2022 Cannes Film Festival. That year, she had a starring role opposite Gilles Lellouche in the action thriller Kompromat, directed by Jérôme Salle. Her part was French- and Russian-speaking, and she worked with dialect coaches in preparation for it. The film was released to mixed reception, and of Kulig's role Paste's Andy Crump wrote: "Diminishing the shine of an actress as luminous as Kulig is an impressive feat, but she's left darkened as Lellouche's object of desire, unable to perform beyond the single dimension given to her."
In 2023, Kulig appeared in Rebecca Miller's romantic comedy She Came to Me alongside Peter Dinklage, Anne Hathaway, Marisa Tomei and Brian d'Arcy James. She was Miller's first choice for the role of a Polish immigrant Magdalena as she regards her "the greatest Polish actress, I know". Despite the mixed reception of the film, Kate Erbland of IndieWire found Kulig's performance "heartbreaking", and David Rooney of The Hollywood Reporter thought that "only Kulig ... registers as a compassionate presence grounded in a modicum of truth." Kulig next starred as a wife of a man transitioning to a woman in socially conservative Poland in Małgorzata Szumowska and Michał Englert's drama Woman Of..., which premiered at the 2023 Venice International Film Festival to critical acclaim. The role earned her a nomination for the Polish Academy Award for Best Supporting Actress. Kulig also had a supporting role in Michael Keaton's 2023 noir thriller Knox Goes Away, and made a guest appearance in Apple TV+ 2024 miniseries Masters of the Air set during World War II. Reviewing the former, Pete Hammond of Deadline Hollywood said that "she makes an impression as Knox's faithful lover". In Masters of the Air, she played a Polish war widow who crosses paths with Major John "Bucky" Egan (Callum Turner) during his stay in London; a fan of director Cary Joji Fukunaga, she accepted the role as she wanted to work with him.