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Proper EMA-balance (balance of kinetic energy, linear momentum and angular momentum), pressure-robustness and Re-semi-robustness (Re: Reynolds number) are three important properties of Navier–Stokes simulations with exactly divergence-free elements. This EMA-balance makes a method conserve kinetic energy, linear momentum and angular momentum in an appropriate sense; pressure-robustness means that the velocity errors are independent of the pressure; Re-semi-robustness means that the constants appearing in the error bounds of kinetic and dissipation energies do not explicitly depend on inverse powers of the viscosity. In this paper, based on the pressure-robust reconstruction framework and certain suggested reconstruction operators in Linke and Merdon [Comput. Methods Appl. Mech. Eng.311 (2016) 304–326], we propose a reconstruction method for a class of non-divergence-free simplicial elements which admits almost all the above properties. The only exception is the energy balance, where kinetic energy should be replaced by a suitably redefined discrete energy. The lowest order case is the Bernardi–Raugel element on general shape-regular meshes. Some numerical comparisons with exactly divergence-free methods, the original pressure-robust reconstruction methods and the EMAC method are provided to confirm our theoretical results. [ABSTRACT FROM AUTHOR]