We re-examine the exact solution of the Einstein perfect-fluid field equations with constant-density equation of state to study the conditions under which negative pressures, or tensions (generated, for instance, at a quark–hadron phase transition), can exist in static and spherically symmetric stars, particularly hybrid stars, which have ordinary, positive pressures in the outermost layers and tensions in the interior. Contrary to intuition, it is found that stars with negative pressures can be in a state of hydrostatic equilibrium, because the gradient of the pressure in the radial direction is positive; this has the effect of making gravity repulsive in the sense that geodesics tend to diverge away from the centre of the star. However, we also find that the field equations can yield such a pressure gradient only if the stellar interior has either a negative mass singularity or negative energy density. Other options for constructing stars with both negative and positive pressures are briefly discussed; simple calculations suggest that anisotropic stress-energies may allow stars with hybrid structure.