By operation I mean binary function from pairs of reals to reals. By diassociative I mean that the subset generated by any two numbers is associative - or, any string composed only of two types of values (like abaab) is unambiguous and needs no parentheses (but with three or more types of values, parentheses may be needed).
I determined yesterday that there is no such operation defined solely as a * b = xa + yb + zab, but I don't know about other more complex definitions.
Update: With some horribly laborious calculations, I found that the set of all commutative and diassociative operations on reals defined x * y = a(x² + y²) + bxy + c(x + y) + d is equivalent to that prior class, due to requiring a to equal 0, and thus they are all associative, so even expanding to that larger set of operations, there are none which are diassociative without being associative.
I feel like there's probably some way, somewhere, to prove that no operation on the reals is diassociative without also being associative - but I don't know how to do it.