Rapid advances in additive manufacturing have kindled widespread interest in the rational design of metamaterials with unique properties over the past decade. However, many applications require multi-physics metamaterials, where multiple properties are simultaneously optimized. This is challenging since different properties, such as mechanical and mass transport properties, typically impose competing requirements on the nano-/micro-/meso-architecture of metamaterials. Here, a parametric metamaterial design strategy that enables independent tuning of the effective permeability and elastic properties is proposed. Hyperbolic tiling theory is applied to devise simple templates, based on which triply periodic minimal surfaces (TPMS) are partitioned into hard and soft regions. Through computational analyses, it is demonstrated how the decoration of hard, soft, and void phases within the TPMS substantially enhances their permeability–elasticity property space and offers high tunability in the elastic properties and anisotropy at constant permeability. Also shown is that this permeability–elasticity balance is well captured using simple scaling laws. To demonstrate the proposed concept through multi-material additive manufacturing of representative specimens is then proceed. The approach, which is generalizable to other designs, offers a route towards multi-physics metamaterials that need to simultaneously carry a load and enable mass transport, such as load-bearing heat exchangers or architected tissue-substituting meta-biomaterials.