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Numerical general relativistic radiative magnetohydrodynamic simulations of accretion discs around a stellar-mass black hole with a luminosity above 0.5 of the Eddington value reveal their stratified, elevated vertical structure. We refer to these thermally stable numerical solutions as puffy discs. Above a dense and geometrically thin core of dimensionless thickness h/r ∼ 0.1, crudely resembling a classic thin accretion disc, a puffed-up, geometrically thick layer of lower density is formed. This puffy layer corresponds to h/r ∼ 1.0, with a very limited dependence of the dimensionless thickness on the mass accretion rate. We discuss the observational properties of puffy discs, particularly the geometrical obscuration of the inner disc by the elevated puffy region at higher observing inclinations, and collimation of the radiation along the accretion disc spin axis, which may explain the apparent super-Eddington luminosity of some X-ray objects. We also present synthetic spectra of puffy discs, and show that they are qualitatively similar to those of a Comptonized thin disc. We demonstrate that the existing xspec spectral fitting models provide good fits to synthetic observations of puffy discs, but cannot correctly recover the input black hole spin. The puffy region remains optically thick to scattering; in its spectral properties, the puffy disc roughly resembles that of a warm corona sandwiching the disc core. We suggest that puffy discs may correspond to X-ray binary systems of luminosities above 0.3 of the Eddington luminosity in the intermediate spectral states.