  Co-orbital planets (in a $1:1$ mean motion resonance) can be formed within a
Laplace resonance chain. Here, we develop a secular model to study the dynamics
of the resonance chain $p:p:p+1$, where the co-orbital pair is in a first-order
mean motion resonance with the outermost third planet. Our model takes into
account tidal dissipation through the use of a Hamiltonian version of the
constant time-lag model, which extends the Hamiltonian formalism of the
point-mass case. We show the existence of several families of equilibria, and
how these equilibria extend to the complete system. In one family, which we
call the main branch, a secular resonance between the libration frequency of
the co-orbitals and the precession frequency of the pericentres has unexpected
dynamical consequences when tidal dissipation is added. We report the existence
of two distinct mechanisms that make co-orbital planets much more stable within
the $p:p:p+1$ resonance chain rather than outside it. The first one is due to
negative real parts of the eigenvalues of the linearised system with tides, in
the region of the secular resonance mentioned above. The second one comes from
non-linear contributions of the vector field and it is due to eccentricity
damping. These two stabilising mechanisms increase the chances of a
still-to-come detection of exoplanets in the co-orbital configuration.
