Phase reduction is a powerful technique that makes possible to describe the dynamics of a weakly perturbed limit-cycle oscillator in terms of its phase. For ensembles of oscillators, a classical example of phase reduction is the derivation of the Kuramoto model from the mean-field complex Ginzburg-Landau equation (MF-CGLE). Still, the Kuramoto model is a first-order phase approximation that displays either full synchronization or incoherence, but none of the nontrivial dynamics of the MF-CGLE. This fact calls for an expansion beyond the first order in the coupling constant. We develop an isochron-based scheme to obtain the second-order phase approximation, which reproduces the weak coupling dynamics of the MF-CGLE. The practicality of our method is evidenced by extending the calculation up to third order. Each new term of the power series expansion contributes with additional higher-order multi-body (i.e. non-pairwise) interactions. This points to intricate multi-body phase interactions as the source of pure collective chaos in the MF-CGLE at moderate coupling. II. MEAN-FIELD COMPLEX GINZBURG-LANDAU EQUATIONThe MF-CGLE consists of N diffusively coupled Stuart-Landau oscillators governed by N coupled (complex-valued)

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The dynamics of an ensemble of N weakly coupled limit-cycle oscillators can be captured by their N phases using standard phase reduction techniques. However, it is a phenomenological fact that all-to-all strongly coupled limit-cycle oscillators may behave as "quasiphase oscillators," evidencing the need of novel reduction strategies. We introduce, here, quasi phase reduction (QPR), a scheme suited for identical oscillators with polar symmetry (λ − ω systems). By applying QPR, we achieve a reduction to N + 2 degrees of freedom: N phase oscillators interacting through one independent complex variable. This "quasi phase model" is asymptotically valid in the neighborhood of incoherent states, irrespective of the coupling strength. The effectiveness of QPR is illustrated in a particular case, an ensemble of Stuart-Landau oscillators, obtaining exact stability boundaries of uniform and nonuniform incoherent states for a variety of couplings. An extension of QPR beyond the neighborhood of incoherence is also explored. Finally, a general QPR model with N + 2M degrees of freedom is obtained for coupling through the first M harmonics.