Despite their different climates, tundras and deserts actually have a few important things in common. They usually receive very little in the way of precipitation, tend to lack large vegetation, and have extreme temperatures. They are both harsh landscapes, and animals there have had to adapt accordingly.
There are no wild animals that can be found in both of these biomes year-round. However, similar conditions, such as extreme temperatures, have led different yet similar species to reside in both places. For instance, burrowing rodents are at home in the desert and in the tundra. Different species of foxes are found in the different biomes and make a living hunting these rodents and other small animals.
However, these species have made their own adaptations in order to survive in their given climate. An arctic fox would easily overheat in most deserts, and a kit fox, which is happy in the deserts of the American Southwest, would never survive the brutal cold of the tundra. Both biomes also have their owls, hawks, wild cats, ruminants, and more. Yet each is adapted to its particular environment.
There are some migratory birds that spend some of their time in deserts and some of their time in the tundra. The arctic tern, for instance, migrates from one polar region to the other over the course of the year. In between, they may stop at desert oases to rest and eat before continuing their long journey. Other migratory birds spend half the year in the tundra and the other half in deserts. Snow geese will nest in the tundra and spend the winter in more southerly latitudes, including the deserts of Mexico and the United States.