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Will you guys stop piling on the LW for complaining about the discount clothes and the water bottles? I suspect she's probably identifying these things as tangible symptoms of something bigger, rather than fixating entirely on them, though I'd be irritated by them too.
LW should try to relax about these things a little. I'm not sure where she's living, but I've seen plenty of American tourists dressed exactly this way in Paris--and yes, I wanted to sink through the sidewalk on their behalf and mine. However, I think she's right about the clothes: dressing well while sightseeing, especially in a foreign country, particularly where you might be visiting religious sites (in Italy, Spain, or France, particularly) is a sign of respect, and one can do so both comfortably and relatively cheaply. My husband buys most of his clothes, which consist mostly of khakis and wrinkle-resistant button-down shirts and short-sleeve polos--at Target, usually on sale. By refusing to dress appropriately--dressing instead as if he's about to go out and play--the LW's brother is showing perverse disrespect for another culture and a lack of maturity. Not to mention that by behaving boorishly, he's representing the United States very poorly, and the LW has every right to be embarrassed about this and to feel that he's also showing disrespect for her, as his host. And if he's rude at home, he's probably not behaving very well in public, either.
While I immediately agreed with Cary and others about the possible cause of his behavior--it sounds exactly like the Asperger's people I've known--as someone else has said, it doesn't matter what's wrong with him; you have to deal with him, no matter what. If you can't get him treated, you can at least find ways to cope with him, and maybe the information on dealing with an adult relative with Asperger's would help. But most of all, find a way to support his wife. She's the one who's bearing the full brunt of this difficult person's behavior.