What do you mean by the "standard kit?" I've never crossed a desert, but given that they comprise rocks as well as sand I would think full body armour is just as essential as it is on the road or track. (Mine has saved me from severe injury plenty of times.) In other words, body armour is "standard kit" in all conditions in my view. It's light, comfortable and cheap and there's really no reason not to wear it.
Broadly speaking I can think of 4 choices - but I'm sure there are more.
1) a well ventilated cordura suit with built-in armour, e.g. the Hein Gericke Tuareg or BMW Rally. With these kind of suits a back protector is often an optional extra - I recommend you buy it. I love my Rally suit - great vents, tough, and with a Camelbak pocket. Great on tarmac as well as dirt.
2) motocross jersey and pants, and a chest protector. Unlike the other choices here,a motocross outfit will give you chest and upper arm protection. But no pockets for wallet etc. And poor protection from abrasions on tarmac.
3) armoured street clothes, e.g. denims or cargo pants by Giali. www.giali.com.
Not as crash proof as a cordura suit. But versatile, and low key. May be supplied with cheap armour without a CE mark. Bin it and replace it with quality armour. Hein Gericke have some lightweight flexible stuff which is easy to fit (no sewing required) and comfortable to wear.
4) A Dainese Safety Jacket and Pants. This is a lightweight mesh armoured undersuit, which you can combine with absolutely anything you like over the top. Wear what the locals wear and blend in if you want to. Or an old army surplus jacket or what have you. Could be a cheap and versatile solution. Have a look at www.sondel.co.uk
and click on Back Pro's.
Hope this helps.
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