This contribution summarizes our proposal on representing microstructures of heterogeneous materials that goes beyond the classical Periodic Unit Cell-based concepts. Inspired by computer graphics [1], we adopt the formal notion of Wang tiles [2] and store microstructural information within a set of domains with predefined mutual compatibility instead of a single cell. Following simple assembly rules similar to jigsaw puzzles, this extension allows for the almost instant generation of arbitrarily large stochastic samples with consistent spatial statistics. We start with a brief introduction of the concept fundamentals and present methods suitable for compressing microstructural information into a set of tiles [3, 4]. Next, we present a reduced-order modeling strategy that exploits the repeating occurrence of individual tiles in generated microstructures. We pre-compute the collective characteristic response of the compressed microstructural representation to a macroscopic loading and extract microstructure-informed modes for the fluctuation part of a macroscopic solution. These modes are then plugged into a macroscopic numerical scheme utilizing a kinematical ansatz of the eXtended Finite Element Method [5]. Using an illustrative 2D elliptic problem, we demonstrate that our scheme delivers less than a 3% error in both the relative L2 and energy norms already with only 0.01% of the unknowns when compared to the fully resolved problem, with further improvements, e.g., by local refinement possible. Acknowledgments. This work received support from the Czech Science Foundation Project No. 19-26143X. [1] Cohen M. F., Shade J., Hiller S., Deussen O. Wang tiles for image and texture generation, ACM Transactions on Graphics, Vol. 22 (3), 287–294, 2003. [2] Wang H., Proving theorems by pattern Recognition – II. Bell System Technical Journal, Vol. 40, 1–41, 1961. [3] Doškářř M, Novák J, Zeman J., Aperiodic compression and reconstruction of real-world material systems based on Wang tiles, Physical Review E, Vol. 90 (6), 062118, 2014. [4] Doškář M, Zeman J., Rypl D., Novák J, Level-set based design of wang tiles for modelling complex microstructures, Computer-Aided Design, Vol. 123, 102827, 2020. [5] Doškář M, Zeman J. Krysl, P., Novák J. Microstructure-informed reduced modes synthesized with Wang tiles and the Generalized Finite Element Method, Computational Mechanics, Vol. 68, 233–253, 2021.