Are there any bread machines that would allow making a sourdough rye bread? If not, could a bread maker simplify the process of making a sourdough loaf?
There are three courses of action for producing sourdough with a bread machine:
Cheater Sourdough Recipe derived from Pain Sur Poolish:
This recipe (with slight modifications) is now my standard bread, which I bake frequently. I'm still tweaking the exact time to bake at the reduced temperature; if the top is browning excessively but the interior is underdone, it may help to flip the loves over for the last few minutes of cooking.
The Zojirushi Home Bakery Supreme can be programmed for rise times up to 24 hours.
It comes preprogrammed to make sourdough, using the "commercial" (type 2) method. You could do the type 1 (traditional) method using its custom settings, especially since you can modify the program as its running (e.g., keep an eye on it, see that its ready after 10 hours, tell it to go ahead and bake now).
Details are in the instruction manual, starting at pp. 24 (PDF page 13).
That said, despite owning one, I personally make sourdough with a stand mixer and an oven. You'll also note the Zojirushi is comparable in price to a good stand mixer.
I often make sourdough with just my starter. After feeding my starter and letting it double in size in the usual way I take what I need for the recipe add everything to the pan and put on a dough cycle. After about five minutes I check to see how wet my dough is. I often have to adjust the dough by adding flour or water a table spoon at a time until the consistency is good. I leave it to mix for five more minutes then cancel the program and restart a normal whole wheat cycle with the delay timer set to give me a total bread making time of 12 hours inclusive of the starter times. For example time for starter to double in size is four hours. Bread machine cycle set for 8 hours. I have produced good sourdough loaves and often get this ready at night so I wake to fresh bread. With a few attempts my timings got better and the bread was light chewy and sour. It's easier than people say and fits well with my shift work.