Tie additional kitchen string to end ties on beef tenderloin. Secure to ends of a wooden spoon long enough to balance on rim of a large Dutch oven. (String should be tied so beef hangs below pot rim and broth but doesnâ??t touch bottom.) Set aside.
Heat oil in Dutch oven over medium heat. Add onions and cook, stirring occasionally, until golden brown and caramelized, about 20 minutes. (If mixture begins to darken or dry out, add a few tablespoons water.) Add wine and cook, stirring and scraping browned bits off bottom, until almost evaporated, about 4 minutes. Add leeks, turnip, rutabaga, and 10 cups water. Bring to a simmer, then partially cover and adjust heat to maintain a bare simmer. Cook 30 minutes, then remove turnip and transfer to a dish. Continue simmering until rutabaga is fork-tender, about 10 minutes more. Remove rutabaga and transfer to dish.
Push leeks and onions to sides of pot and lower beef into center, balancing wooden spoon on pot rim. If necessary, add more warm water to completely cover beef. Add potatoes and bring to a simmer. Cook, adjusting heat as necessary to maintain broth temperature at 190 degrees, until a thermometer inserted in center of beef reads 120 degrees for rare, about 30 minutes. (Meat will continue to cook after poaching.)
Remove potatoes and transfer to dish with vegetables. Remove beef, transfer to a cutting board, and let rest, tented with foil, 5 minutes. Keep poaching liquid warm. Remove strings from beef and cut beef into slices. Strain poaching liquid through a fine-mesh sieve. Divide broth, vegetables, and meat among serving bowls. Season with salt and pepper and top with prepared horseradish.