  The $C^3$ approach is an invariant formalism that utilizes the eigenvalues of
the Riemann curvature tensor to match spacetimes across a specific matching
surface. We apply this approach to match an anisotropic fluid with an exterior
vacuum solution, including the case in which discontinuities appear on the
matching surface. As a particular example, a class of analytic solutions, which
describe the gravitational field of realistic neutron stars, is matched to the
exterior Schwarzschild spacetime.
