Step 1: Lock onto the buzzwords - $sulphur\ granules$ plus sinus tracts plus an anaerobe equals one disease: actinomycosis.
Step 2: The organism is $Actinomyces\ israelii$, a branching gram-positive filamentous bacterium that grows anaerobically/microaerophilically and is NOT acid-fast.
Step 3: Those yellow granules are tangled colonies of the bacterium wrapped in a polysaccharide-protein shield that blocks phagocytosis, which is why the infection becomes chronic with discharging sinuses.
Step 4: Rule out distractors using oxygen and acid-fastness - Nocardia is aerobic and partially acid-fast (no true sulphur granules), while Salmonella and the tularemia agent Francisella are unrelated systemic pathogens.
\[\boxed{\text{Actinomyces}}\]