Heat olive oil in a stockpot over medium-high heat; saute garlic until browned on all sides, 3 to 4 minutes. Transfer garlic to a work surface, reserving oil in the stockpot. Slice garlic.
Mix rice in the reserved oil until evenly coated; add tomatoes and cook, stirring frequently, until toasted and fragrant, about 10 minutes. Stir beef broth and spinach into tomato mixture; cook and stir for 5 minutes.
Stir garlic into spinach mixture, cover stockpot, and simmer soup until flavors combine, 15 to 20 minutes. Season soup with salt and pepper.