Dissolve the yeast in the warm water in a large bowl and let rest 5 minutes, until foamy.
Add the first clear flour, rye flour, rye sour starter, salt, and caraway seeds to the bowl; stir with a wooden spoon until combined.
Turn the dough onto a well-floured surface. With a wet hand and a bowl scraper or dough cutter, knead by scraping, folding, pulling and stretching for about 5 minutes until the dough is elastic and resists stretching.
If the dough feels insufficiently elastic, add first clear flour 1/4 cup at a time and continue kneading, up to 3 more minutes, keeping the dough fairly soft.
Transfer the dough to a clean, wet bowl; keep your hands wet and pat the dough, then cover lightly with a film of water and plastic wrap. Place in a warm spot and let rise until doubled, about 45 to 60 minutes.
Prepare a baking sheet by sprinkling with a mixture of cornmeal and rye flour.
Wet a clean work surface and your hands. Divide the dough in half and gently shape each portion into a round, handling gently so it does not tear. Let rest up to 10 minutes; no further proofing is required.
Place an empty broiler pan on the oven floor and preheat the oven to 375°F for 5 minutes.
Brush the loaves with water and place them on the prepared baking sheet on the middle rack (or on a preheated stone). Carefully add 6–8 ice cubes or 1 cup hot water to the broiler pan on the oven floor and quickly close the door to create steam.
Bake 5 minutes with steam, then remove the broiler pan. Use a skewer to make 10–12 holes around each loaf's crust, return to the oven, reduce heat to 350°F, and bake 10 minutes.
Make holes again and brush the loaves with water. Continue baking until the crust is hard and the loaves sound hollow when tapped on the bottom, about up to 1 hour total (about 50 minutes in the original test).
If desired, transfer loaves to a preheated stone and bake an additional 10–15 minutes for a crisper crust.
Remove the bread from the oven, brush the tops with more water, and cool on a wire rack before slicing.