A taxing seven- or eight-day wilderness trek crosses from the Manali Hwy at Rumtse to Korzok over several very high passes. You'll need to carry all your own food and camping equipment, so most trekkers hire horses and a guide. A shorter alternative (four to five days) starts from the west side of Tso Kar over the Horlamkong La (4712m) and three other passes, but remains very taxing.
An excellent trek is the six-night, seven-day walk from Tso Moriri to Kibber in Spiti (Himachal Pradesh) along an ancient trade route. Most of the hike is in high-altitude rolling valleys but there is one major pass (5500m) with some glacier walking. Day three starts with wading a river that can become uncrossable in the afternoon. Foreigners can only do the trek from north to south, as a Tso Moriri permit is needed from Leh.
Horses (₹450) and guides (₹1000) are available in Korzok given a day or two's warning.
Climbing options from Korzok include 6151m Mentok Kangri, an adventurous trekking peak-style alternative to popular Stok Kangri. You need to be fully self-supported. A climbing permit is required from Leh.