As the title says, what modification is needed, so a creature cannot fart while still being able to excrete their waste?
So creatures that can't excrete or doesn't have an anus like the Demodex is out of this question.
Is this possible?
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The modification to prevent an organism from eructating or breaking wind (thus, avoiding the f**t word) is simple. Its microbiome would have to be highly efficient and was capable of consuming metabolic products that would tend to be gases. Therefore, its alimentary system wouldn't be producing gases in sufficient qualities for them to be expelled from its fundament. It would still excrete matter, but this would be in the form of solids and liquids.
I wonder if this creature may have some kind of gas absorbing gland or organ that prevents it before excretion or something - possibly even has another organ to release it in a different way - probably soundlessly (so maybe something that looks a little like a vent/gills?)