Step 1: Trace the offending molecule. Gut bacteria turn dietary choline and related compounds into trimethylamine, a volatile substance with a strong fishy smell.
Step 2: The liver normally detoxifies it by N-oxidation into odourless trimethylamine oxide, a reaction performed by flavin-containing monooxygenase 3, written $FMO3$.
Step 3: If $FMO3$ is deficient, as in trimethylaminuria or fish odour syndrome, unconverted trimethylamine builds up and is shed in urine, sweat and breath, explaining the characteristic odour.
Step 4: Dismiss the others: fumarylacetoacetate hydrolase belongs to tyrosine breakdown, methane monooxygenase is microbial, and D-amino acid oxidase has no role here.
\[\boxed{\text{Flavin-containing monooxygenase 3 (FMO3)}}\]