The limitations of the human body, of clothing, and of hygiene and cosmetic products inevitably arise in any dance where you spend 10-12 minutes at a time hugging another person, in a crowded room that as often as not lacks air conditioning. I suppose there are the privileged few who don't sweat much, or who always manage to dance in places with good air circulation and/or air conditioning, but that isn't most of us, most of the time. There are things one can control:
• Wear not merely a deodorant but an antiperspirant – and not just on your armpits, but also on the interior of your elbows, on your wrists just above the hand, and anywhere else you sweat. (Yeah, yeah, it's carcinogenic and you'll die.)
• Don't wear makeup, because even the best rubs off even if you believe you never put your face near your partner (you do . . . you might not realize it, but you do).
• Ditto hairspray. In addition to rubbing off, hair that's fixed in place makes you hotter and sweatier, and much of it triggers allergic reactions in others (as does most perfume). If you have lots of hair, simply tie it up or back. I'm a man with exceptionally long hair, but at the milonga, I look like a man with very short hair because I tie it.
• Wear clothing made from light natural fibers, not synthetics that don't breathe, and definitely not velvet (which is horrific in tango). I prefer bamboo, others swear by linen or cotton, but anything like polyester is deadly. "Technical cloth" is an exception, in that it's often synthetic but physically constructed to breathe, but I've yet to find attractive clothing made of it; I don't want to dance dressed like a hiker or a basketball player.
• Some people claim that undershirts, vests/waistcoats, or jackets create an absorbent layer, or at least keep sweat from reaching their partner. Your mileage may vary on that one. Me, if I wear one, two, or all three of those with a shirt, I get dangerously overheated and sweat so much from my head that my partner gets a literal, visible sweat shower. Not a good choice for me. I prefer just a shirt, and I'm not too proud to unbutton it to a near-1970s-disco extent to maximize cooling. What can I say . . . works for me.
• Maintain good head posture to minimize face contact. You won't avoid it, however much you think you will. I work very hard on this yet end up with makeup-covered faces and hairspray-laden hair smooshed against my face because my partners who supposedly don't like face contact make unconscious contact, including one who stopped dancing with me because she found it too sweaty . . . who literally mashes her face into every leader (yes, we gossip).
• Embrace lower than the upper back, perhaps at waist level. Heat rises, even in the body, and the higher your embrace, the more heat it traps to cause sweating. The very romantic "hand on the neck" thing is, well, very romantic . . . and hot as hell, given that many people sweat profusely from the neck.
• Take pauses to drink cool water and lower your body temperature, ideally near a fan or open window.
• Change out of sweaty clothing. Quite a few men change shirts. I usually go through 3 or 4. Women seem less inclined to change, but I know several who do, and it makes a big difference. One woman in particular makes a game of having an outfit for early, mid, and late milonga.
All of which points to "simpler is better." People who coat their face and hair in various products, wear complicated outfits with layers or strange fabrics, and can't change because their desired look can't be achieved during a 10-minute break . . . well, they're asking to be hot and sweaty.
"Simple" doesn't mean "inelegant" or "ugly." I get compliments on my outfit quite often – including from fellow leaders who have no ulterior motives, like "Dance with me!" – and my typical outfit consists of a solid-colored bamboo shirt (no tie, vest, or jacket), proper tango trousers with a slightly expensive belt, and tango shoes kept in reasonable shape. My life and dance partner wears zero makeup, hair products, or perfume, and usually sticks to a camisole-style top with spaghetti straps, made for tango, and a short skirt made for tango, both in a solid color, plus her shoes (which, I grant, are fancy). Other followers grab her by the arm to ask where she buys her clothing.
People overestimate what it takes to achieve simple elegance.