Furuncle or Carbuncle: A large, red, tender boil with mucopurulent discharge is most commonly caused by a bacterial infection, typically Staphylococcus aureus, leading to the formation of a furuncle (boil) or carbuncle (a group of connected boils). The size and symptoms described are consistent with this diagnosis.

Abscess: An abscess is a collection of pus that has built up within the tissue of the body, which could present similarly to a large boil. The presence of mucopurulent discharge supports this possibility.
Infected Cyst: An infected sebaceous cyst or epidermoid cyst could present as a red, tender, and swollen lesion with discharge, similar to a boil.
Cellulitis: While cellulitis is an infection of the skin and subcutaneous tissues, a localized, tender, and red area could be mistaken for a large boil, especially if there's a focal point of infection.

Necrotizing Fasciitis: Although less common, necrotizing fasciitis is a severe infection that can start with symptoms similar to a boil but rapidly progresses to severe pain, swelling, and skin necrosis. It's critical to identify early due to its high mortality rate.
Gas Gangrene: Caused by Clostridium perfringens, gas gangrene can present with severe pain, swelling, and crepitus (a grating or bubbling feeling under the skin), which might initially resemble a large, infected boil.

Actinomycosis: A rare bacterial infection that can cause abscesses or boils, especially in the face and neck area, but can occur anywhere on the body.
Mycobacterial Infections: Certain mycobacteria, like those causing tuberculosis or leprosy, can lead to skin lesions that might resemble boils, though they are much less common and typically have a more chronic course.
Deep Fungal Infections: Fungal infections like sporotrichosis can cause skin lesions that might initially look like boils, especially in immunocompromised individuals or after exposure to contaminated soil.