The majority of tandems are setup with what can best be described as bomb proof wheels. Given the extra weight and torsional forces placed on the spokes, rim, and hubs a decent set will include 36 count 15 gauge spokes, appropriately matched to heavy gauge hubs and rims.
Even the best made wheels will eventually fail. The most susceptible to premature failure is the rear wheel. Signs of premature wear include nicked spokes (from chain slap or kicked up stones), slop in the bearings, or dented rims. It is best to repair or replace parts as damage occurs but occasionally failure will occur during rides.
9 times out of 10 this failure will be limited to a broken spoke. Spokes are tightened under extreme loads causing sudden failure. Although startling this failure is almost never catastrophic. Usually unnoticeable except for the metallic “ping” of the spoke snapping.
A failed spoke will immediately throw a wheel out of true. Tandems equipped with disc brakes will be okay to limp home while bikes with rim brakes will need some truing to continue on. To nurse the tandem home first find a safe place to stop.
Depending on the spoke that broke and where the break occurred you will likely need to secure the spoke from flailing around and damaging another component. If the spoke can not be removed carefully bend and wrap it around a nearby spoke to secure its movement.
After securing or removing the broken spoke use a spoke wrench to adjust neighboring spoke tension to re-align the wheel. Do not expect to get the wheel perfectly back to true just straight enough to get your team home. Re-mount and continue on careful not to place too much strain on the weakened wheel.
The strong wheels associated with tandems should allow your team to return home safely. Remove the wheel and stop by your local bicycle shop to purchase a replacement spoke and have the entire wheel inspected for further signs of impending failure. A broken spoke, though startling, is never usually a ride-ender for tandems.