There are several historical sources about what and how people in the Dhofar region ate in the past; here I would like to highlight a few authors.
(the photo by Salwa Hubais is of bidah, the bulb from the white flower, gladiolus candidus, which is cooked and eaten)
Bent, James. “Exploration of the Frankincense Country, Southern Arabia.” The Geographical Journal 6.2, 1895, 109-33.
Bent, James and Mabel Bent. Southern Arabia. London: Elibron,  2005.
Bent, Mabel. The Travel Chronicles of Mrs. J. Theodore Bent, Volume III: Deserts of Vast Eternity, Southern Arabia and Persia. Gerald Brisch, ed. London: Archaeopress, 2010.
Theodore and Mabel Bent were in the Dhofar region from December 20, 1893 until January 23, 1894. During their short stay, they traveled along the coast and a short distance in the mountains; I believe they are the first Westerners to visit the Dhofar mountains and write a description of it. In Southern Arabia (1900/ 2005), they recount that along the coast they saw coconut palms, “bright green fields,” “[t]obacco, cotton, Indian corn, and various species of grain” (233); as well as gardens with “the plantain, the papya, mulberries, melons, chilis, brinjols [eggplants], and fruits and vegetables of various descriptions” (234). During their journey through the mountains, they saw the still-used method of cooking meat on heated stones (250); plant-life including sycamores, acacia, jessamine, convolvulus, maidenhair ferns and fig trees (256); and describe how rice was eaten (275). They also met an elderly sheikh who had 500 head of cattle and 70 camels (250).
Thomas, Bertram. Arabia Felix: Across the Empty Quarter of Arabia. Jonathan Cape: London 1932. reprint.
In Arabia Felix (1932) Thomas recounts his journey across the Rub al Kahli (Empty Quarter) in the fall/ winter of 1930. His trip started in Salalah, so there are a few food references. For example, at the home of a prosperous merchant, he is given a meal of “beef grilled crisp and black, spaghetti drenched in tomato sauce, and slices of pineapple” (19). During a short trip in the mountains, he explains that the Gibali diet was milk, honey and beef (51) and that hyena, fox and “eggs, chicken and all manner of birds are under strict taboo” (59). He also discusses the custom of killing half of a man’s cows at his death (55-56).
Janzen, Jorg. Nomads in the Sultanate of Oman: Tradition and Development in Dhofar. London: Westview Press, 1986.
Janzen, who did his fieldwork from January to August 1977 and January to May 1978 (xxi), identifies nine bands of vegetation: the coastline belt, grassland of coastal plain, bush and tree vegetation of the foothills and escarpment, grassland of the lower and middle levels of the plateau, bush and tree vegetation of the mountain wadi area, grassland of the upper plateau, desert vegetation of the transition zone to the Nejd, desert vegetation of the Negd and sand desert vegetation (34-35). He notes that there “are many indications that the plateaus were once more thickly wooded than they are now” and that the “last stands of trees” on the coastal plain were cut down in the 1960s (35).
He discusses the traditional “monsoon-rain fields” in which millet and beans are grown along with cucumbers, tobacco, maize, “red” (chili) peppers, and tomatoes in the mountains (107, drawing 106, details of planting 108). In the mid-1950s, when diesel pumps could bring up water faster and cheaper than animal labor, crops included millet, wheat, maize, “watermelons, cucumbers, tomatoes, sweet potatoes, eggplants, onions and peppers,” as well as bananas, papayas and coconuts (154). In the mid-1970s, the composition of plantations changed in that cereals were no longer sown given that corn and flour became easily available and that land was given over to fruits and vegetables which were in higher demand (154).
His book also includes important historical data such as balance of trade data for 1896 which lists the top six exports: incense, butter fat (samn), cotton, skins, latex, sharks’ fins; and top six imports: rice, sugar, cotton cloth, dates, coffee, wheat (47) and a chart on the “Movement of Livestock Prices in Dhofar” for cattle, camels and goats with data from 1965-70, 1971-75 ad 1976-78, showing, for example, the cost of a milk cow as 40-100 OR in 1971-75 and 250-330 OR in 1976-78 (102).