Eating less meat is associated with a healthier body and planet. Yet, we remain reluctant to switch to a plant-based diet, largely due to the sensory experience of plant-based meat. The gold standard test to analyze the texture of food is the double mechanical compression test, but it only characterizes one-dimensional behavior. Here we use tension, compression, and shear tests along with a constitutive neural network to automatically characterize the mechanics of eight plant- and animal-based meats across the entire three-dimensional spectrum. We discover that plant-based sausage and hotdog, with stiffnesses from 35.3kPa to 106.3kPa, successfully mimic the behavior of their animal counterparts, with stiffnesses from 26.8kPa to 115.5kPa, while tofurky with 167.9kPa to 224.5kPa is twice as stiff, and tofu with 22.3kPa to 34.0kPa is twice as soft. Strikingly, the more processed the product--with more additives and ingredients--the more complex the mechanics: The best model for the softest, simplest, and oldest product, plain tofu, is the simplest, the classical neo Hooke model; the best model for the stiffest products, tofurky and plant sausage, is the popular Mooney Rivlin model; the best models for all highly processed real meat products are more complex with quadratic and exponential terms. Interestingly, all animal products are stiffer in tension than in compression, while all plant-based products, except for extra firm tofu, are stiffer in compression. Our results suggest that probing the fully three-dimensional mechanics of plant- and animal-based meats is critical to understand subtle differences in texture that may result in a different perception of taste. We anticipate our models to be a first step towards using generative artificial intelligence to scientifically reverse-engineer formulas for plant-based meat products with customer-friendly tunable properties.