Improving naturalness in Gauge Mediation with non-unified messenger sectors

We study models of gauge-mediated supersymmetry breaking with messengers that do not belong to complete representations of grand-unified gauge groups. We show that certain setups characterized by heavy Wino can greatly improve the fine tuning with respect to models with unified messengers, such as minimal gauge mediation. The typical models with low tuning feature multi-TeV superparticles, with the exception of the Higgsinos and possibly Bino and right-handed sleptons. As a consequence, the absence of signals for supersymmetry at the LHC is trivially accommodated in our framework. On the other hand, testing these models will be challenging at the LHC. We finally show that the gravitino can be a consistent candidate for cold dark matter, provided a rather low reheating temperature, if a standard thermal history of the universe is assumed.