In what way did the works of Marx and Engels influence Levi-Strauss in writing Tristes Tropiques and, in a wider sense, his structural analysis of social phenomena?
Levi-Strauss said that he had three "mistresses": Marxism, (Freudian) Psychoanalysis, and Geology. So Marx clearly played a prominent role in his thinking, as he did for many intellectuals, especially in France, in the mid-twentieth century. That said, Levi-Strauss's primary debt to Marx was not direct. Marx had claimed (or implied) that to understand any given human institution, idea, or historical development, one had to understand the material underpinnings of it. People's lives were determined, or "structured," by economic realities and social class. Religion, ideology, politics, and so on were "superstructural," not explicable in their own terms, but rather by the economic structures they reflected. Crucially, this was a universal condition. All culture, wherever it was found, was a reflection, in a sense, of economic foundations.
As Levi-Strauss said in Tristes Tropiques, Marx "established that social science is no more founded on events than physics is founded on sense data." He illustrated the importance of theoretical models in social science. Like the other two "mistresses," Levi-Strauss tells us, Marx also demonstrated the importance of looking below the surface for explanations of behavior:
All three demonstrate that understanding consists in reducing one type of reality to another; that the true reality is never the most obvious; and that the nature of truth is already indicated by the care it takes to remain elusive.
Levi-Strauss thought that ideologies, culture, and language were expressions of structures within the human mind. These structures were universal, not, as some had argued, different between "primitive" and "modern" societies. Their expression in the form of practice was shaped by surroundings and circumstances. This idea was laid out elsewhere in Levi-Strauss's work, but is present in Tristes Tropiques as well, where he explicitly compares the myths and religious beliefs of peoples in Brazil with others as far away as India.