  We investigate the clustering transition undergone by an exemplary random
constraint satisfaction problem, the bicoloring of $k$-uniform random
hypergraphs, when its solutions are weighted non-uniformly, with a soft
interaction between variables belonging to distinct hyperedges. We show that
the threshold $\alpha_{\rm d}(k)$ for the transition can be further increased
with respect to a restricted interaction within the hyperedges, and perform an
asymptotic expansion of $\alpha_{\rm d}(k)$ in the large $k$ limit. We find
that $\alpha_{\rm d}(k) = \frac{2^{k-1}}{k}(\ln k + \ln \ln k + \gamma_{\rm d}
+ o(1))$, where the constant $\gamma_{\rm d}$ is strictly larger than for the
uniform measure over solutions.
