This is not even a hypothetical feature. There is at least one language which used a type information to resolve associativity [1]. I have even used one, it works, but was very hard to get accustomed for me.
[1] https://dl.acm.org/doi/10.1145/1596655.1596667 (especially search for "juxtaposition")
Looking closer, it's not as bad as what I had in mind. Association order only depends on type, not value (which is what I meant by data). This means you can figure it out at compile time, and actually makes some sense if you want to be able to use standard mathematical notation like they do, which has operators with different association orders.
The interpreter takes a brute-force approach of going through each interpretation until it finds one that works. Sometimes, it simply hangs forever, and other times, it interprets the program differently than you'd expect. But I always considered that as part of the charm of golfing in it.