What causes the degrees of freedom to be halved for Majorana fermions?

In many textbooks and on this site (nanophys answer here) it is stated that 'Majorana Spinors have half the degrees of freedom of a typical Dirac spinor'. A generic spinor in 3+1D has 8 degrees of freedom, from 4 complex components. The Dirac Equation reduces this to 4 degrees of freedom.

However since the Majorana spinor is 'real' does it have half the degrees of freedom because it's components cannot be complex, or because it must be left invariant under the charge conjugation operator? Or are these, in the Majorana basis, the same condition?