How does a “break” in the neutral wire enable it to reach the full line voltage

It sounds like any current taken by an appliance in the shed is traveling thru the live #6 wire and returns via the soil back to the main box at your home. This is unacceptable in the UK according to wiring regulations.

The reason it would be unsatisfactory is because earth (and earth wires) are reserved for enabling the detection of faults (should one occur). Imagine you have a hand operated drill that developed a live wire fault to the metal on the drill - you'd be electrocuted and there would be nothing to protect you - any current passing through you to earth is undetectable from actual appliance current.

Many, many places insist on RCDs (residual current devices) - these wire in series with live and neutral and any current that does not flow through both live and neutral, generates an imbalance current that can be detected and trip the supply. This usually means you will receive up to 30mA for a few tens of milli seconds maximum - short enough to prevent you frying.

Another problem of not having a neutral and relying on earth return thru soil is that if the soil dries you get no decent connectivity.

If you want to avoid this do some math on what #6 wire can safely withstand and rewire one of the two wires making it a neutral wire.

If you have totally balanced loading on your two phases you will not be able to detect any change. EDIT: Note that this is pretty much never going to be the case.

1. EDIT: If you have a poor ground connection or the utility provider has a poor ground connection you will have a floating ground. Unless the neutral is bonded to a good local ground point it will float. It will float in potential closer to the phase that has the heavier 115V loads. If you have a 10A load on one phase and a 1A load on the other phase you may find that the 1A 115V load sees 200V and the 10A 115V load sees only 30V. Electronic and reactive devices will behave unpredictably at 200V or 30V if they are designed for 115V. Resistive loads will dissipate more or less depending on the supply voltage, over voltage will easily burn out transformers, globes and motors, under voltage will also affect electronic power supplies and synchronous motors that fail to get to speed.
2. They can will be sharing the utility neutral until the point where the neutrals is are bonded in the fuse cabinet. From this point it would be unwise to share it for reasons like you describe but I can see it happening when extra outlets are added with a mix of 115V and 230V without due supervision. EDIT: The consumer is not expected to cope with a floating utility side neutral or ground, it should never occur, in some wiring codes they are bonded in the fuze box, in others the ground is local and the neutral comes from the utility transformer common connected to the ground at the utilities transformer. Sharing an local neutral conductor between unbalanced loads is bad practice and not conformant to most wiring codes.

EDIT:

Below you can see how two dissimilar loads will cause the floating neutral point to be pulled from the earth reference if they share a neutral wire and it is compromised. In this example with the loads selected the larger load will be running at undervoltage (23V) and may survive if it is an incandescent lamp or heater but may fail if it is a motor or electronic load. The other load will see 207V and will likely suffer some failure no matter what sort of load it is.

It is also worth noting that the floating neutral point will be 92V above the protective earth, if any, in this example.

^{simulate this circuit – Schematic created using CircuitLab}