It happened IOTL: it's called the "Union of Utrecht" or "Union of Old Catholic Churches", and the split happened after Vatican I mostly on the issue of papal infallibility (there are other doctrinal issues). The official date for the slit is 1889. The UU is not in communion with Rome, but is in communion with the Anglican Church. Their approach is much more liberal than the Roman church's (celibacy is optional, women have been ordained and same-sex unions are blessed in church) and obviously they believe that the church doctrine should only be decided by church councils.
The UU is (sadly) quite small nowadays (the faithfuls are estimated to be around 90,000). It would be however possible to change things: there was serious opposition to the doctrine of papal infallibility in 1869 (mainly spearheaded by German theologians but also supported by major figures in the Italian church, including the archbishop of Bologna. A number of priests and bishops). Let's assume that dissent is a bit stronger and the council becomes polarized, so that there is no vote on papal infallibility. When Italy takes Rome in September 1870, Pius IX decides to leave the city (the possibility was discussed IOTL) and call the second round of Vatican I in France or Spain (OTL there were proposals to call a second round either in Trento or in Malta but both the British and the Austrians were very much against it). The second round will result in a stronger declaration of papal infallibility (with excommunications raining around) but the result of a deep and acrimonious split of the church will be there for anyone to see. ITTL the "Old Catholics" will be supported by national governments (certainly in Italy and in Germany, where Bismarck will see an opportunity for his idea of kulturkampf. Note that the doctrine of papal infallibility was not well received in Austria either, where the government wanted to keep a firm hold on the church). This will not happen immediately, but over the years I believe that there is a very good chance for the Old Catholics to become very well established and their numbers a century later might be really impressive.