Infrared imaging of the colliding-wind binary Apep has revealed a spectacular dust plume with complicated internal dynamics that challenges standard colliding-wind binary physics. Such challenges can be potentially resolved if a rapidly-rotating Wolf-Rayet star is located at the heart of the system, implicating Apep as a Galactic progenitor system to long-duration gamma-ray bursts. One of the difficulties in interpreting the dynamics of Apep is that the spectral composition of the stars in the system was unclear. Here we present visual to near-infrared spectra that demonstrate that the central component of Apep is composed of two classical Wolf-Rayet stars of carbon- (WC8) and nitrogen-sequence (WN4-6b) subtypes. We argue that such an assignment represents the strongest case of a classical WR+WR binary system in the Milky Way. The terminal line-of-sight wind velocities of the WC8 and WN4-6b stars are measured to be $2100 pm 200$ and $3500 pm 100$ km s$^{-1}$, respectively. If the mass-loss rate of the two stars are typical for their spectral class, the momentum ratio of the colliding winds is expected to be $approx$ 0.4. Since the expansion velocity of the dust plume is significantly smaller than either of the measured terminal velocities, we explore the suggestion that one of the Wolf-Rayet winds is anisotropic. We can recover a shock-compressed wind velocity consistent with the observed dust expansion velocity if the WC8 star produces a significantly slow equatorial wind with a velocity of $approx$530 km s$^{-1}$. Such slow wind speeds can be driven by near-critical rotation of a Wolf-Rayet star.
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