I think I might occasionally repost some of the answers I posted there, to here, if only because I sometimes want to refer back to answers I gave and this blog moves (a lot) slower than Reddit.
Today, I just answered the question: can bacteria in the small intestine cause breath to stink?
To which I answered:
"In the case of people with "healthy" gastrointestinal tracts, these sphincters are one of the ways by which we keep the proper direction of flow (of food, which later becomes faeces along with all the associated gases), which in turn (usually) keeps bacteria where they're supposed to be.
The fact that the food that leaves the stomach (in a kind of watery paste called chyme) is still very acidic means it's still relatively inhospitable for bacteria, and has had less time for bacteria to enter/grow, which is why the small intestine ordinarily has much less bacteria (fewer species and fewer cells) than the large intestine (which is where the digesting stuff actually becomes faeces). Peristalsis (the movement of your intenstines that pushes everything along) drives the flow, making sure faeces and flatulence goes out the right hole. This movement is why you can still sometimes fart even when constipated (unless you're really impacted).
All these mechanisms keep the 'lower gases' coming out of the mouth - which is why in the vast majority of cases of bad breath originates from bacteria in the mouth. However IBS and constipation are obviously cases where all the digestive stuff isn't working like normal.
People with severe IBS and/or constipation can sometimes get so backed up (past the valve that you mentioned) that the small intestine gets way more bacteria then normal - this is called small intestine bacterial overgrowth (SIBO). In this situation, gases produced by bacteria in the small intestine can work their way up and out through the mouth (presumably bubbling up through the stomach as its sphincters open and close to let food through/burps out), including hydrogen sulphide, which is one of the classic halitosis related molecules. Checking someone's breath for volatile compounds produced by bacteria is actually one of the major ways by which doctors diagnose this condition.
I should say, I am not a gastroenterologist so I don't know whether these volatiles are frequently present at levels detectable to the human nose, which isn't something that seems to get measured a lot in the scientific literature, however I did find a couple of non-peer-reviewed reports like this that say bad breath can be a symptom of SIBO. In fact, some people even think that SIBO might be a/the cause of IBS (because if you treat it, you can reduce IBS symptoms in some people), but this is contentious and still being worked on.
So to answer your general question - it's plausible, although without testing it's impossible for anyone to say whether it's actually the case for you. Generally maintaining good oral hygiene is the best recipe to prevent bad breath, but in some cases there might be deeper medical issues, so it might be worth seeking the advice of a gastroenterologist."