Antibiotic wouldn’t have stopped the bloodclot.
The death certificate will read as follows.
Primary Cause of Death: Pulmonary embolus
**** Secondary to: Deep vein thrombosis of the calf
**** Secondary to: Severe cellulitis of the calf
**** Secondary to: Unhygenic tattoo procedure
A Primary Cause of death is what actually killed you.
The Secondary Causes of death are all the factors that brought about the primary cause.
Dirty tatto => Severe cellulitis => Deep vein thrombosis => Pulmonary embolus => Death
Dirty tatto + Early antibiotics => No severe cellulitis => No deep vein thrombosis => No pulmonary embolus => No death