Add the watermelon and cucumber to a food processor or blender, working in batches if needed. Blend until smooth — about 30 seconds in a food processor.
Set a mesh strainer over a large bowl. Pour the watermelon and cucumber puree through the strainer to remove the seeds. Press down gently to extract as much liquid as possible — don’t worry if some pulp seeps through.
Add the pepper, cilantro, almonds, jalapeno, and about 1 cup of your strained liquid back into the food processor/blender. Blend until a smooth pesto forms — about 1 minute. Stir the pesto into the reserved soup base. Whisk in the olive oil and lime juice. Season with black pepper.
Chill at least 1 hour or until ready to serve.
If desired, garnish with additional diced watermelon and chopped cilantro. Allow each diner to season their bowl with bitters or sherry peppers sauce.