Summary: Quasi-local variables, i.e. quantities whose values can be derived from physics accessible within an arbitrarily small neighborhood of a spacetime point, are used to construct the equation of state (EoS) for the anisotropic fluid in the spherical symmetry. One parameter families of equilibrium solutions are obtained making it possible to assess stability properties by means of the standard \(M(R)\) method. Normal modes of radial pulsation are computed as well and are found to confirm the onset of instability as predicted by the \(M(R)\) method. As an example, a stable configuration with outwardly increasing energy density in the core is obtained with a simple quasi-local extension of the polytropic EoS. It is also found that the loss of stability occurs at higher surface compactness when the anisotropy of pressures is present.