It is not everyday that we get to tour an actual desert while on horseback. You do not have to go to places as far as the Sahara to experience this. Chile also has deserts! It is called the Atacama Desert, which can be found in Northern Chile.
Hidden Trails offers expeditions through this desert and beyond that will expose you to what was it like during the 11th century in Chile, as the trip also include a visit to the Andean Highlands. Too bad you won't be riding in a camel.