  We propose a real-space formalism of the topological Euler class, which
characterizes the fragile topology of two-dimensional systems with real wave
functions. This real-space description is characterized by local Euler markers
whose macroscopic average coincides with the Euler number, and it applies
equally well to periodic and open boundary conditions for both crystals and
noncrystalline systems. We validate this by diagnosing topological phase
transitions in clean and disordered crystalline systems with the reality
endowed by the space-time inversion symmetry $\mathcal{I}_{ST}$. Furthermore,
we demonstrated the topological Euler phases in quasicrystals and even in
amorphous lattices lacking any spatial symmetries. Our work not only provides a
local characterization of the fragile topology but also significantly extends
its territory beyond $\mathcal{I}_{ST}$-symmetric crystalline materials.
