Coupled atmosphere-fire modeling is recognized as a relevant approach for the representation of the interaction between a wildland fire and local meteorology at landscape scales. The atmospheric model component used in the coupled system is based on several approximations, which are adopted for computational efficiency or physical processes representation, including the widely-used anelastic approximation. The validity domain of the anelastic approximation may be questioned in the context of high-resolution wildland fire modeling due to the large fire-induced heat releases near the surface. This study aims to study this question with the MesoNH anelastic model coupled with the Blaze fire model. A compressible version of the MesoNH-Blaze coupled model has been developed for comparison with the anelastic system. The FireFlux I experimental fire is used for this comparative study conducted at a 10-m and a 25-m horizontal atmospheric resolution. Results show significant anelastic/compressible differences at a 10-m resolution on the physical processes occurring near the fire with higher horizontal velocities and the presence of gravity waves downstream of the fire. This is in addition to the fire plume with realistic larger vertical velocities. Differences at a 25-m resolution are much smaller in all evaluated processes. The compressible system only enriches the physics underlying fire-atmosphere interactions at a very high resolution, which means that the anelastic approximation remains relevant for large-scale coupled atmosphere-fire simulations, considering the significant economy concerning numerical costs.