Born in the provincial Cuban town of Sagua La Grande in 1902, Lam was the son of a Chinese shopkeeper and a Cuban woman of Spanish and African parentage. His maternal grandmother had been a Congolese slave and his godmother, Ma’Antonica Wilson, was a priestess in the Afro-Cuban religion Santeria, in which traditional rituals, forbidden during the period of slavery, were kept alive behind a veil of conventional Catholicism.

Wilson believed Lam had the sensitivity to become a seer, but Eskil Lam, the artist’s son, says his father was not interested. “He was already intent on becoming an artist, which in Cuba at that time meant imitating European models,” he says. Lam showed such promise at Havana’s art school that his home town provided him with a scholarship to study in Madrid. He arrived in 1923, aged 21, began studying under the conservative academic painter Fernando Alvarez de Sotomayor, and set about forging a career as a romantic portrait painter.

The death of his first wife Eva and infant son of tuberculosis in 1931 convinced him Spain was not an equal society – if he had more money, his family would have had better medical treatment – and pushed him towards socialism and experimentation in painting.

While recovering in Barcelona during the Spanish Civil War, in which he fought on the Republican side, he met a friend of Picasso, who gave him a letter of introduction. Lam fled to Paris when Barcelona fell to Franco’s forces and summoned up the courage to visit the great painter. Picasso welcomed Lam warmly and, as they talked, he handled an African mask resting on a nearby table. “My father was transfixed by the appearance of the mask and its rocking movement on the tabletop,” says Eskil Lam. “Picasso noticed this and said: ‘You should be proud of your ancestors.’ My father was baffled and asked why. And Picasso said: ‘Because they made this.’ ”

It was Lam’s first sighting of African art and his first perception of himself as connected to Africa as an artist. Yet the strong relationship that developed between the two artists led to the perception in some quarters of Lam as a mere Cuban imitator of Picasso.

“Picasso was more of an enabler to my father than a direct influence,” says Eskil Lam. “He allowed him to feel good about who he was and where he came from.” Lam’s Paris career was taking off when he found himself on the run again in the face of the German assault on Paris in 1940 – heading south on the last train out of the Gare d’Orsay, which was hit by enemy aircraft fire. In Marseille, he got a place on a cargo boat heading to America, packed with writers, artists and intellectuals, including the surrealist leader André Breton and the anthropologist Claude Levi-Strauss – part of the great wartime brain-drain.

It feels hugely symbolic that after a tortuous voyage of a month and another lengthy delay in Martinique, Lam, the sole non-white member of this great modernist caravan was denied entry to the US; rather than outright racism, the probable reason was that as a citizen of a neutral country, Cuba, he could not be given refugee status.

His return to Cuba after 18 years abroad seemed a dismal exile from the place – Europe – where he had achieved so much. To Lam, it was an island mired in poverty and corruption, where degraded versions of Afro-Cuban rituals were trotted out for the amusement of American tourists.

“It was like some sort of hell,” he told an interviewer years later. “For me, trafficking in the dignity of a people is just that: hell. I refused to paint cha-cha-cha.

“I wanted with all my heart to paint the drama of my country but by thoroughly expressing the Negro spirit, the beauty of the [visual] art of the blacks.”

The result was Lam’s most distinctive, powerful and personal paintings, suffused with references to Afro-Cuban mythology: cubistic portrayals of altars to African deities (known as “Orishas”) evoking a world of sacrifice and trance possession; mask-like faces peering from tropical forests where human, animal and plant forms merge.

Lam declared that he wanted to be a kind of Trojan horse for Cuban culture in the Western art world, who would “spew forth hallucinatory images with the power to surprise, to disturb the dreams of the exploiters”.

Yet Lam’s attitude to this world of ancestral belief remained highly ambivalent. His principle informant on Afro-Cuban culture wasn’t his godmother, the Santeria priestess, but Lydia Cabrera, an upper-class, white anthropologist.

In Haiti, he attended a voodoo trance ceremony – not as a “local” who had observed similar rituals as a child, but as part of a group of European intellectuals, including Breton. “A lot of people have assumed we lived in a Santeria environment,” says Eskil Lam. “Nothing could be further from the truth. We had the most atheist upbringing you can imagine. My father lost his faith in any kind of religion in the late Twenties and he never deviated from that.”

Nonetheless, only someone intimately connected to this metaphysical world could, you feel, have produced works of such numinous intensity.

After the Second World War, Lam lost no time in leaving Cuba. With his path to America still blocked, he divorced his second wife, Helena Holzer, who had settled in New York, and returned to Paris in 1952, later creating a home in Albissola, in Italy, with his third wife, Lou Laurin.

He didn’t entirely neglect Cuba, however, producing major murals for public buildings in Havana. With the Cuban revolution of 1959, Lam became a kind of informal cultural ambassador for the Castro regime, hosting an international writers’ conference in Havana in the euphoric, “revolutionary” year of 1968 – despite the fact that he wasn’t actually living in Cuba at the time. “He realised things in Cuba weren’t as rosy as some people claimed,” says Eskil Lam. “But the world was so polarised at that time. If you were against Castro, you were with the hotheads in Miami calling for an American invasion, and he was never going to be part of that.”

Lam died in 1982, yet his story raises questions of personal and cultural identity that feel more relevant now than ever.

We live in an era of cultural hybridity, but also, paradoxically, a time when borders are being guarded jealously – whether through Europe’s anxious response to global migration or the absurd ideologies of Islamic fundamentalists. It’s a moment when questions of who has the right to be what and to be where feel ever more pressing.

“If you look at those great artists who fled to America before and during the Second World War, of whom my father was one,” says Eskil Lam, “they weren’t asking to be seen as ‘Europeans’ but to be accepted as modern artists. That was the way my father thought. He aimed for universality as a human being, to be a citizen of the world.”

Now, more than ever, that feels like an ideal worth pursuing.

The EY Exhibition: Wifredo Lam is at Tate Modern, London, from Sept 14 to Jan 8 2017;  tate.org.uk