  Spectroscopic properties of low-lying states and cluster structures in
$^{12}$C are analyzed in a "beyond mean-field framework" based on global energy
density functionals (EDFs). To build symmetry-conserving collective states,
axially-symmetric and reflection-asymmetric solutions of the relativistic
Hartree-Bogoliubov equations are first projected onto good values of angular
momentum, particle number, and parity. Configuration mixing is implemented
using the generator coordinate method formalism. It is shown that such a global
microscopic approach, based on a relativistic EDF, can account for the main
spectroscopic features of $^{12}$C, including the ground-state and linear-chain
bands as well as, to a certain approximation, the excitation energy of the
Hoyle state. The calculated form factors reproduce reasonably well the
available experimental values, and display an accuracy comparable to that of
dedicated microscopic cluster models.
