In sociology, a homosocial relationship is one between two people of the same sex which is not of a sexual or romantic nature. Most people these days are familiar with the word "bromance," which means much the same thing. The homosocial relationship at the heart of "The Beach of Falesá" is that between Wiltshire and Case. I would like to suggest that it is directly related to the imperialist theme that Robert Louis Stevenson wishes to convey in the story.
It is interesting to note that Stevenson conceived "The Beach of Falesá" as an example of literary realism. He sought to portray life in the South Sea islands with complete fidelity to the reality of everyday life, warts and all. Among other things, this meant showing, without prejudice, the effects of British colonial rule upon the indigenous population.
The relationship between Wiltshire and Case is arguably one that could not have happened had they remained in England. Their personalities are different, for one thing. Yet, their destinies collide in the setting of a remote tropical outpost of the British Empire, the South Sea island of Falesá. The empire has brought them together. Their homosociality is a product of colonialism and their respective roles in it. Despite their differences in personality and temperament, the two men are brought together by shared ties of race and culture. Naturally, they gravitate toward each other.
Wiltshire and Case are traders. As such, they have come to Falesá to exploit the opportunities presented to them by colonial commerce. As well as race and culture, then, they are drawn together by economic necessity. Though they are rivals competing for the same business, they do at least have a mutual understanding.
Despite the ostensible bond between them, the homosocial relationship between Wiltshire and Case is fraught from the outset. As their relationship emerged out of a social context of fraud, exploitation, and cutthroat competition, it is not surprising that their connection develops along similar lines. The fateful acquaintance of the two men is a microcosm of what is happening in the world around them.
For all of Case's superficial bonhomie and good will, he is too experienced at the game of colonialism not to spot an opportunity for domination and control when he sees one. Case starts to treat Wiltshire the same way he treats every native he has ever come across: a sucker to be cheated and exploited. He is also a master at using superstition to his advantage, a practice to which even white colonialists like Wiltshire are not immune.
When Wiltshire realizes he has been had, his relationship with Case is effectively over. The bonds between them have been irreversibly severed. Additionally, Wiltshire's love for Uma has given him a new perspective on life, one no longer dominated by colonial preconceptions and prejudices. His connection to Case was purely skin deep, based as it was on superficial similarities of race, gender, and occupation. In throwing off the shackles of homosociality, Wiltshire has embraced his humanity and cast off his accretions, which led to his being suckered into Case's cruel world of dark dishonesty.