  We solve completely the spectral synthesis problem for reproducing kernels in
the de Branges spaces $\mathcal{H}(E)$. Namely, we describe the de Branges
spaces $\mathcal{H}(E)$ such that all $M$-bases of reproducing kernels (i.e.,
complete and minimal systems $\{k_\lambda\}_{\lambda\in\Lambda}$ with complete
biorthogonal $\{g_\lambda\}_{\lambda\in\Lambda}$) are strong $M$-bases (i.e.,
every mixed system $\{k_\lambda\}_{\lambda\in\Lambda\setminus\tilde \Lambda}
\cup\{g_\lambda\}_{\lambda\in \tilde \Lambda}$ is also complete). Surprisingly
this property takes place only for two essentially different classes of de
Branges spaces: spaces with finite spectral measure and spaces which are
isomorphic to Fock-type spaces of entire functions. The first class goes back
to de Branges himself, the second class appeared in a recent work of A.
Borichev and Yu. Lyubarskii. Moreover, we are able to give a complete
characterisation of this second class in terms of the spectral data for
$\mathcal{H}(E)$. In addition, we obtain some results about possible
codimension of mixed systems for a fixed de Branges space $\mathcal{H}(E)$, and
prove that any minimal system of reproducing kernels in $\mathcal{H}(E)$ is
contained in an exact system of reproducing kernels.
