The phrase "desert under fire" can be replaced with a number of synonyms based on context. For example, "burning desert" conveys a sense of intense heat and danger. "War-torn desert" highlights the conflict and turmoil in the region. "Bombed-out desert" speaks to the devastating impact of military attacks. "Hostile desert" emphasizes the danger posed by natural elements such as extreme temperatures, rugged terrain, and lack of resources. "Barren battlefield" draws attention to the aftermath of war and destruction. Ultimately, the choice of synonym depends on the situation being described and the desired emphasis or tone of the writing.