While most breads made from white or whole wheat flour are high in FODMAPs, traditionally made sourdough breads made with these flours are low. The difference lies in the way the bread dough is handled. Most store-bought breads have baker's yeast added to promote rapid fermentation. The gases produced in the fermentation process cause the bread to rise quickly so it is ready to bake as soon as possible.
Sourdough bread, on the other hand, does not rely on the addition of bakers yeast. (Clue: authentically made sourdough bread usually does not include baker's yeast on the list of ingredients.) Instead, a “starter” saved from the previous batch of bread provides a complex mixture of yeasts and lactic-acid producing bacteria to ferment the dough. The fibers in the bread dough are in contact with the starter for as much as two days, allowing plenty of time for the microbes in the starter to digest most of the FODMAPs for you. Eventually the bread rises sufficiently for baking. Breads made with the sourdough process have varying degrees of that characteristic tangy taste; try several varieties to find the one you like best.
I'm learning how to make sourdough bread from scratch, and when I've nailed it, I'll share my secrets with you! For now, I've got to buy it at the store. I've been asked to recommend some brands of sourdough bread. Unfortunately, my recommendations will be of limited value unless you live in Maine, because sourdough breads are almost always made without preservatives. That means they are not going to turn up in the bread aisle at your grocery store. They tend to be locally made and sold directly from the bakery, or shipped daily to nearby grocery stores. To complicate matters, they don't all say "sourdough" on the label. Sourdough refers more to the bread-making process than to the flavor or variety of the bread. A few especially awesome grocery stores bake their own authentic sourdough breads--ask to speak to the head baker to find out which ones qualify. Hannaford makes a Monterey boule; Whole Foods make pan au levain. Both of these stores also resell the local favorite, Borealis Breads, who make Aroostook Wheat, French Peasant and Italian loaves. (Notice none of the bread names include the word sourdough. Borealis comes with a food label, which refers only to "starter," including wild yeast--that's OK and indicates an authentic sourdough process bread). I've heard that Wegman's makes a mean sourdough.
Please use the comments to share your local favorite brand of authentically made sourdough bread along with where you buy it.
Update below 1/20/2017