Combine all the ingredients in a large pot and bring to a boil over high heat. Reduce to a simmer and cook partially covered for 1 hour.
Take the broth off the heat and let it cool a little before straining. To strain, place a fine mesh sieve over a bowl and pour the broth over it. You should have about 6 cups.
Transfer the vegetable broth to mason jars and let it come to room temperature before covering and storing in the refrigerator or freezer.
Combine the corn, halved tomatoes, onion, garlic, vinegar, and salt together in a large bowl and mix together. Let it sit for 30 minutes to 4 hours at room temperature. The longer the mixture sits, the more seasoned it will be.
Pour into a blender and blend on high speed until smooth. Transfer to a large mixing bowl.
In the blender, add the broth, olive oil, and beans. Blend on high speed until smooth. Slowly pour into the large mixing bowl along with the corn and tomato mixture.
Stir in the lime juice and season with salt to taste. Serve cold.
The Corn Cob Vegetable Broth is optional. Adding corn to vegetable broth causes it to come out slightly sweet and it helps add more of that sweet corn flavor to this gazpacho.
*Butter beans are mature lima beans and are a beige/white color. Most brands in the U.S. just call them Butter Beans so you shouldn't have a problem but occasionally regional differences will occur and the term "butter beans" will be applied to both green and white lima beans. You'll want to use the mature white lima beans.