For about twenty years, the question about the essential factors promoting the long-lasting coexistence of trees and grasses in humid savannas is at the center of several mathematical works, by the construction of deterministic and/or stochastic mathematical models. A closely related topic is coexistence of open savanna and forest patches at landscape scales, which raises the challenge of accounting for contrasted spatial patterns under similar climate conditions through fire mediated tree-grass interaction models. In this work, we propose and study a deterministic spatio-temporal fire-mediated tree-grass interactions model. The model is based on two nonlocal reaction-diffusion equations with kernels of intra and inter-specific interactions, corresponding to woody and grassy biomasses. A novelty in this paper is the consideration of a kernel-based nonlocal facilitation of trees by other trees to promote growth of seedlings/shrubs and, indirectly, limit fire propagation and its impact. We also take into account a kernel-based nonlocal competition of trees on grasses for light availability and nutrients. A qualitative analysis of the model is carried out and it reveals several ecological thresholds that shape the overall dynamics of the system. Depending on these thresholds, monostability of the forest, grassland or savanna space-homogeneous stationary state and multistabilities (i.e. involving more that one space-homogeneous stationary state) are proven possible. Thanks to the nonlocal biomasses interactions, our model accounts for the occurrence of space inhomogeneous solutions, including a possibly periodic spatial structuring sometimes observed in the humid savanna zone. Specifically, linear stability analyses, performed in the vicinity of space-homogeneous stationary states, provides conditions for the appearance of space inhomogeneous solutions including spatially periodic or aperiodic ones. Finally, numerical simulations are presented to illustrate our theoretical results. Notably, we verify that the computed spatial wavelengths were in good agreement with the predictions from the theoretical analysis.