We propose a novel asymmetric spin wave interferometer to detect ultralight axion dark matter. The axion modulates spin-wave properties via a weak effective magnetic field in ferromagnets. The interferometer splits a spin-wave source into two paths of different lengths and sets them to interfere destructively. The system then converts the axion-induced phase shift into a measurable magnetization oscillation that can radiate electromagnetic waves and generate electrical signals via Faraday induction. The signal-to-noise ratios have been evaluated for three detection schemes: the linear amplifier, the single-photon detector, and the electrical signal detection approach, accounting for both magnetization fluctuation and thermal noise. The accessible axion mass range is approximately 10⁻⁸ eV to 10⁻⁶ eV, set by the spin wave propagation length and the relaxation time.