Manuka honey (not regular honey) has been found to inhibit the growth of h. pylori
(
http://www.medscape.com/viewarticle/458834_5), and there are reviewers of the product on Amazon and elsewhere who say that it has worked for them.
Despite initial skepticism, I had a good experience with it myself, not with h. pylori (as far as I know), but with candida. You would think such a sugary product would be the worst possible thing for candida; I can only say it seemed to work despite that.
However, I found that it MUST be used with probiotics, which should be taken later in the day, or its antimicrobial effect creates imbalances. And it should be treated like a medicine, not a food; 1 to 2 teaspoons a day, no more.
It comes in different strengths; you'd want the higher ones. Wedderspoon is an organic brand commonly available around here (Berkeley); you'd want the 16+, not the 12+.