Halfway through my conversation with the Korean film-maker Bong Joon-ho, I notice a tattoo peeking out from under the sleeve of his jacket. “It means ‘wife and son’,” he says, gesturing at the inky characters on his wrist. “Sorry to disappoint. But I do have this …” With that, he yanks down the neck of his T-shirt to reveal a detailed illustration of a tree; its brown branches and green leaves extend across one corner of his chest, over his shoulder, on to his back and out of sight. He and his cinematographer got matching tattoos when they made the 2009 psychological drama Mother, about a woman fighting to help her son beat a murder rap. “There was a beautiful tree on location so we did this to commemorate the shoot.” Glancing down at it now, he wrinkles his nose. “It looks a bit like a skin disease.”

Were he to commemorate his latest movie, Okja, in a similar fashion, I suggest that the Netflix logo might be an appropriate choice after the controversy caused in Cannes by the film, which was financed by the streaming giant. “Maybe next time,” he replies with a rueful smile.

Okja is Bong’s sixth feature, and the third in a series of bold and bonkers fantasies that have led Quentin Tarantino to compare him to Spielberg in his pomp. Each one contains a chilling ecological or environmental warning. In The Host, the dumping of chemicals in the Han river results in a giant rampaging monster equally content to devour people in water or on land. The spectacular Snowpiercer is confined to a high-speed train that circumnavigates the globe continuously, populated by the last inhabitants of the frozen earth. Meanwhile, Okja concerns the friendship between a young Korean girl and a giant manatee-style creature, genetically engineered to provide the planet with cheap meat. Bong crams a lot into two hours. There is an eccentric cast, including his Snowpiercer star Tilda Swinton and Jake Gyllenhaal as a zany TV presenter in safari shorts. The tones range from pastoral lyricism to grisly satire. And the director’s pet themes– man-made disaster, class war, the menace of capitalism – all get a look-in.

The 47-year-old director was taken aback by the intensity of dissent in Cannes over Netflix. “I thought maybe there would be an academic seminar: ‘What is the future of film?’ Something like that.” Objections centred on the fact that Okja and the festival’s other Netflix production, Noah Baumbach’s The Meyerowitz Stories, were bypassing French cinemas due to the country’s insistence on a three-year window between theatrical release and streaming. The jury president Pedro Almodóvar said he couldn’t imagine a film winning the Palme d’Or without guaranteed French distribution (sure enough, the Netflix titles went home empty-handed), while the festival changed its rules to exclude in future any competition titles that would not be screened in cinemas. Bong seems a touch dazed by it all. “If only Noah Baumbach and I had studied French law,” he jokes.

Other studios, he claims, were reluctant to support Okja. “Netflix wasn’t. If you want to make something strange, it is a good place to go.” The project originated in the image of an exotic creature that popped into his head four years ago. He grabs a tablet and starts scrolling through the images on the screen, tilting it toward me when he finds what he’s looking for: his original sketch of Okja herself. She is larger than the cinematic version and more canine in essence, though with all her lumbering cuteness intact. The film itself might have been purely whimsical had Bong not emphasised the story’s harsher elements, including a sequence set in a slaughterhouse. “Films either show animals as soulmates or else we see them in documentaries being butchered. I wanted to merge those worlds. The division makes us comfortable but the reality is that they are the same animal.”

Bong’s mother ruled out cinema visits throughout his childhood in Daegu and Seoul. “She was obsessed with hygiene. She thought theatres were infested with germs because they never saw any sunlight.” I point out that he has chosen to make his living in an art form that his mother associates with grime and infestation. “Also,” he says, warming to the theme of parental discord, “she refuses to talk about my film Mother.” He has assured her that she wasn’t the inspiration for that picture’s domineering main character, who tries to poison her son when he is a child and then mollycoddles him as an adult, spooning soup into his mouth while he pees against a wall. “All my other films we speak about. But that one she never mentions.” I remind him that his father also took against his black comedy Barking Dogs Never Bite because the disagreeable main character, a university professor, had the same job as him. So his movies have managed to offend both parents. He grins. “I don’t care.”

In all other respects, a typical Bong film tends to bring people together, overriding any cultural and linguistic boundaries. In short, they travel well. “My movies are based in genre, which is a universal language. Everybody speaks it.” The one glaring communication breakdown in his career occurred when the Weinstein Company bought Snowpiercer. Harvey Weinstein hacked 25 minutes out of the two-hour movie, only relenting when the test scores for the original cut were far higher than ones for his sliced-and-diced version. I wonder, then, if Bong was thinking of anyone in particular when he wrote the scene in Okja in which one of the creatures is carved up. “That really wasn’t about Harvey,” he protests. There is, however, a genuine reference to the Snowpiercer debacle buried in Okja. “When one character talks about an underground laboratory where terrible things happen, he says it’s in Paramus.” This, he explains with a twinkle, is the New Jersey town where the Weinstein Company held the test screening for their cut of Snowpiercer.

Jake Gyllenhaal in Okja. Photograph: Barry Wetcher/Jae Hyuk Lee/Netflix

That movie ended up with a limited US release and still has yet to open in Britain. “Only Harvey knows why,” Bong says. But it lives on: a TV series based on the film and starring Jennifer Connelly is in production, with Bong and his compatriot Park Chan-wook, director of Oldboy and The Handmaiden, as executive producers. And he assures me there is no bad blood between him and Weinstein. They even bumped into each other this year in Cannes. “It was pleasant. He congratulated me on Okja. I don’t hate him. It’s just his style; it’s what he does. Whereas my style is final cut.”

Besides, there are bigger things to worry about than Weinstein. Most of Bong’s films focus on the battle between unlikely, working-class outsiders and the venal corporations and governments who try to crush them. “All our problems arise because of capitalism. It brings pleasure but also so much pain and unhappiness. The questions I ask in my films about why we harm the environment or animals all come down in the end to capitalism.”

The train in Snowpiercer, with the rich living it up in the luxurious front carriages and the poor subsisting in squalor at the back, is a fast-moving metaphor for that very system. “The train explodes at the end. The survivors must go outside. Now they are free from the train but we don’t know what will happen to them.” So is he saying that this is what society should do, that we should blow up the train? “Yes!” he whoops excitedly. “And then we all go outside and freeze together and die!”

Okja is on limited cinema release and available on Netflix from 28 June.