We demonstrate the existence of broadband anomalous transmission of electromagnetic (EM) waves through an electrically very narrow aperture in a thin metal plate backed by left-handed media (LHM). It is shown that the incoming energy is simply squeezed through the aperture, without any reflection, regardless of how small the physical aperture is. Analogous to the Venturi effect in fluids, the EM energy behaves as an incompressible fluid, flowing through the constriction (aperture), as if the wave character of EM was lost. This counterintuitive tunnelling effect is not related to resonances and is impossible to achieve with naturally occurring materials. Analysis indicates that a cascaded waveguide implementation, in the form of a slotted metallized LHM wall, retains the exotic broadband transparency character. Applications for a device with these properties abound in the RF/microwave/THz/optical ranges, to which the analysis can be scaled. To complement this analysis, a proposal for achieving broadband LHM is also included.