Must a classical Lagrangian or a Hamiltonian be a real function?

$\bullet$ Is it fair to assume that the classical Hamiltonian or Lagrangian of a system (a particle or a field) is always a real-valued function?

$\bullet$ If not, can you provide counter-examples?

$\bullet$ The cases in which the Hamiltonian represents the total energy of a system, it must be real on physical grounds. Apart from that is there any other criterion which demands that a Lagrangian or Hamiltonian must be real?