We propose an experimental realisation of the protocol for the counterfactual disembodied transport of an unknown qubit---or what we call counterportation---where sender and receiver, remarkably, exchange no particles. We employ cavity quantum electrodynamics, estimating resources for beating the classical fidelity limit---except, unlike teleportation, no pre-shared entanglement nor classical communication are required. Our approach is multiple orders of magnitude more efficient in terms of physical resources than previously proposed implementation, paving the way for a demonstration using existing imperfect devices. Surprisingly, while such communication is intuitively explained in terms of ``interaction-free'' measurement and the Zeno effect, we show that neither is necessary, with far-reaching implications in support of an underlying physical reality. We go on to characterise an explanatory framework for counterportation starting from constructor theory: local wormholes. Conversely, a counterportation experiment demonstrating the traversability of space, by means of what is essentially a 2-qubit exchange-free quantum computer, can point to the existence in the lab of such traversable wormholes.