  We propose a new generalization of the Yang-Baxter equation, where the
R-matrix depends on cluster $y$-variables in addition to the spectral
parameters. We point out that we can construct solutions to this new equation
from the recently-found correspondence between Yang-Baxter equations and
supersymmetric gauge theories. The $S^2$ partition function of a certain 2d
$\mathcal{N}=(2,2)$ quiver gauge theory gives an R-matrix, whereas its FI
parameters can be identified with the cluster $y$-variables.
