It has been recently pointed out that the unavoidable tuning among supersymmetric parameters required to raise the Higgs boson mass beyond its experimental limit opens up new avenues for dealing with the so called mu-B-mu problem of gauge mediation. In fact, it allows for accommodating, with no further parameter tuning, large values of B-mu and of the other Higgs-sector soft masses, as predicted in models where both mu and B-mu are generated at one-loop order. This class of models, called Lopsided Gauge Mediation, offers an interesting alternative to conventional gauge mediation and is characterized by a strikingly different phenomenology, with light higgsinos, very large Higgs pseudoscalar mass, and moderately light sleptons. We discuss general parametric relations involving the fine-tuning of the model and various observables such as the chargino mass and the value of tan beta. We build an explicit model and we study the constraints coming from LEP and Tevatron. We show that in spite of new interactions between the Higgs and the messenger super fields, the theory can remain perturbative up to very large scales, thus retaining gauge coupling unification.