I'm just wildly spitballing here, but...
Things go even worse for the Qing in the 19th century than OTL- the rebellions (Taiping, Nian, Miao, Panthay, Dungan etc.) are more successful and any semblance of central authority collapses.
Then you could get Brits and others nibbling away from their positions on the coast- perhaps first just as a means of securing their coastal footholds from the anarchy going on, but things could snowball a little. And the Russians would be sticking their noses further into Mongolia and Xinjiang.
Much of the country would remain in the hands of native regimes, but you'd get Europeans involved in their internal affairs- playing different warlords against each other, providing military equipment, seeking advantageous trade conditions etc. You'd presumably get Brits and Russians competing for influence in the westernmost regions of China, with the Great Game and all that- we saw an embryonic form of this in their contacts with Yaqub Beg IOTL.
If you squint hard enough, the Europeans' relationship with native Chinese leaders- at least the ones in their immediate vicinity- might start to very vaguely resemble indirect rule a la Africa and India.