This paper proposes an algorithm for the design of entropy-constrained unrestricted polar quantizer (ECUPQ) for bivariate circularly symmetric sources. The algorithm is globally optimal for the class of ECUPQs with magnitude quantizer thresholds confined to a finite set. The optimization problem is formulated as the minimization of a weighted sum of the distortion and entropy and the proposed solution is based on modeling the problem as a minimum-weight path problem in a certain weighted directed acyclic graph. The proposed algorithm enables solving the overall problem in
$O(K^2\log \hat{\mathcal{|P|}})$ time, where $K$ is the size of the set of possible magnitude thresholds and $\hat{\mathcal{P}}$ is the set of the number of phase levels for the uniform phase quantizers.