What is the likely diagnosis for a 10-month-old female infant with a history of egg and peanut allergies, who presented with fever, vomiting with mucus, and gastrointestinal upset after consuming 4oz of tempeh?

Food Allergy Reaction: Given the patient's history of egg and peanut allergies, and the recent consumption of tempeh (a fermented soybean product), a food allergy reaction is the most likely diagnosis. The symptoms of upset stomach, fever, and mucus vomiting are consistent with an allergic reaction.

Viral Gastroenteritis: The symptoms of fever, vomiting, and upset stomach could also be indicative of a viral gastroenteritis, which is a common illness in infants.
Bacterial Gastroenteritis: Although less likely, bacterial gastroenteritis is also a possibility, especially if the tempeh was contaminated with bacteria.
Soy Allergy: As tempeh is a soy product, a soy allergy is also a possible diagnosis, especially given the patient's history of other food allergies.

Septicemia: Although unlikely, septicemia is a life-threatening condition that requires immediate attention. The presence of fever and vomiting could be indicative of septicemia, especially if the patient's condition worsens.
Intussusception: This is a rare but serious condition where a part of the intestine telescopes into another part, causing bowel obstruction. The symptoms of vomiting and abdominal pain could be indicative of intussusception.
Appendicitis: Although rare in infants, appendicitis is a life-threatening condition that requires immediate surgical attention. The symptoms of fever, vomiting, and abdominal pain could be indicative of appendicitis.

Food Protein-Induced Enterocolitis Syndrome (FPIES): This is a rare but serious condition where the immune system reacts to certain food proteins, causing inflammation and damage to the intestinal lining.
Eosinophilic Gastroenteritis: This is a rare condition where the immune system reacts to certain foods, causing inflammation and damage to the gastrointestinal tract.