  There have been several combinatorial constructions of universally positive
bases in cluster algebras, and these same combinatorial objects play a crucial
role in the known proofs of the famous positivity conjecture for cluster
algebras. The greedy basis was constructed in rank $2$ by Lee-Li-Zelevinsky
using compatible pairs on Dyck paths. The theta basis, introduced by
Gross-Hacking-Keel-Kontsevich, has elements expressed as a sum over broken
lines on scattering diagrams. It was shown by
Cheung-Gross-Muller-Musiker-Rupel-Stella-Williams that these bases coincide in
rank $2$ via algebraic methods, and they posed the open problem of giving a
combinatorial proof by constructing a (weighted) bijection between compatible
pairs and broken lines.
  We construct a quantum-weighted bijection between compatible pairs and broken
lines for the quantum type $A_2$ and the quantum Kronecker cluster algebras. By
specializing the quantum parameter, this handles the problem of Cheung et al.
for skew-symmetric cluster algebras of finite and affine type. For cluster
monomials in skew-symmetric rank-$2$ cluster algebras, we construct a
quantum-weighted bijection between positive compatible pairs (which comprise
almost all compatible pairs) and broken lines of negative angular momentum.
