Texture is fundamental to high-fidelity rendering of 3D digital assets, directly influencing scene detail and visual realism. Existing methods typically adopt 2D texture mapping, where texture images are either manually created or synthesized from exemplars. While advances in texture synthesis have improved 2D texture quality, 2D representations remain inadequate for modeling volumetric meso-structure textures with complex geometry. Methods targeting meso-structure textures often struggle to capture high-frequency details and lack real-time rendering capabilities, limiting their practical use. We propose MesoSplats, a neural implicit method for extracting and synthesizing meso-structure textures using 3D Gaussian splatting. Given the multi-view images containing the meso-structure geometric details, our approach supports texture extraction, synthesis, and real-time rendering. We introduce a mesh-Gaussian hybrid representation that decouples geometry into a coarse base mesh and embedded 3D Gaussians, guided by initial point cloud constraints to enhance reconstruction fidelity. Local implicit texture features are sampled from the base mesh surface and further refined through a proposed Consistency Tuning strategy, which enforces alignment between the reconstruction and sampling spaces. To boost texture synthesis quality, we incorporate a tileability-aware patch-matching algorithm alongside a smoothness regularization on the latent feature space to ensure spatial coherence. Extensive quantitative and qualitative experiments demonstrate the effectiveness of our method.