When lead rope soloing, back-feeding (or auto-feeding) is when the weight of the rope between the anchor and soloist (the live rope) pulls extra rope through the soloist's belay device, causing a dangerous amount of slack to develop.
What can I do to avoid this dangerous situation?
Preventing backfeeding is one of the many dangers in rope soloing that must be mitigated. It is especially pernicious as it is unique to rope soloing, often catching new soloists---who might otherwise be very experienced climbers---by complete surprise. While there are many methods that have been devised, none are perfect and all introduce some extra amount of fiddling into the system. Backfeeding is also a reason that you should always tie backup knots! At worst, the rope will feed through until hitting a backup knot. While not ideal, it is much better to suddenly have an extra 5m of slack instead of 30m.
The first step to preventing backfeeding is knowing when to expect it. In order for backfeeding to occur, the amount of weight on the live rope side must overcome the weight of the dead rope ("brake hand" side) plus the friction through the belay device. A slippery and skinny rope might start backfeeding through a Silent Partner after only 15m of overhanging climbing; a fat and fuzzy rope might never backfeed through an unmodified GriGri on a wandering slab pitch. You should do lots of backyard testing with your intended device and rope to get a feel for how prone your setup is to backfeeding. Additionally, take measures to minimize backfeeding whenever you have a convenient stance on a pitch and especially so before any difficult or consequential sections of a climb. It is much easier to deal with backfeeding at comfortable locations. Experiment until you have a good feel for how often you will need to deploy countermeasures on any given pitch.
All of the following methods can work, but have various pros and cons. Some also require bringing extra equipment along and can't be improvised.
A rebelay is simply connecting your live rope to an anchor partway up a pitch with, e.g., an alpine butterfly. This anchor should be capable of an upward pull (bolt, cam in horizontal crack, opposing pieces), but is still backed up by the original anchor at the base of the pitch. Some soloists advocate clipping rebelays with a long sling (or a knot with a very large loop) so that the force of a fall will be transmitted to the original anchor rather than an upward pull at the rebelay (account for rope stretch!).
In normal climbing, rope drag is the enemy of the leader. While rope soloing, however, rope drag is your friend! By increasing the amount of rope drag, you reduce the effective weight of the live rope and thus the likelihood of backfeeding. As a general rule, the only times you should extend a piece while rope soloing are when routing the rope around sharp edges and when you are worried about a placement walking. Otherwise, use the shortest quickdraws you own, clip directly to the thumb loops of cams, alternate placements between parallel cracks, etc. Intentionally z-clipping several pieces can be almost as effective as a rebelay.
Part of the equation for backfeeding to happen is the amount of weight on the dead rope side. By having a longer loop of rope between the device and the first clipped backup knot, you tip the scales towards not having backfeeding occur.
After clipping a piece of protection, tie a bulky knot in the live rope above this piece. The weight of the live rope is thus held by this knot being unable to pass down through the carabiner. Note that any distance between this knot and the carabiner will be extra slack in the event of a fall. This method works best with small carabiners and fat ropes. Common knots include an alpine butterfly, a slip knot (one hand!), or a clove hitch on a spare carabiner. In the event of a fall, this knot simply gets tightened in the middle of nowhere (or gets pulled free if using a slipknot) and simply keeps transmitting forces on down the rope.
A common solution is to tie a friction hitch (prusik, kleimhesit, french prusik, etc.) around the live rope and clip it to a piece of protection. The subtlety of this method is that we only want this friction hitch to hold the weight of the live rope---not the forces of a fall. This is generally accomplished by one of several methods: 1) clipping the friction hitch to a "fuse" that will break in a fall (fishing line, thread, rubber band, toy carabiner, ...); 2) tying a very weak friction hitch (e.g., single-wrap prusik) that won't bite down in a fall, but just barely holds the weight of the rope; 3) clipping the friction hitch with a long (120cm+) sling so that the hitch won't see any of the forces of a fall, even with rope stretch. Regardless of the method, be sure to pull up any extra slack that might be lurking after you've tied the friction hitch.
Clip a small, stout rubber band to the carabiner, clip the rope, then clip the rubber band again, capturing the rope (N.B.: make sure the rope is actually through the carabiner). Longer rubber bands might take a wrap around the rope for proper tension. Rubber bands for parachute rigging work very well.
After clipping a piece of gear, use a small spring-loaded clamp from the hardware store to pinch the rope against the spine side of the carabiner. I prefer the plastic ones due to a (perhaps irrational) fear of a metal one damaging the rope. Additionally, creating a "basket" between the jaws with a loop of duct tape or similar helps locate the clamp + rope + carabiner. Some people tie a keeper cord to the clamp on the outside chance that it falls off. Binder clips, bread bag clips, and similar have been used to various degrees of success.
The industrious rope soloist ArcticBastard designed what he calls the "Gromm hitch": a 3D-printed plastic doohickey that is preinstalled on the carabiner that pinches the rope. See the youtube "user manual" or the free STL file for 3D-printing. This is by far the fastest method for dealing with backfeeding.
Regardless of how you choose to mitigate backfeeding, the biggest danger lurking is accidentally creating a factor 2 fall scenario. You should also consider the affect your method has on fall dynamics---are you suddenly falling directly on a friction hitch?
When aid climbing or when free climbing on easy/ledgy terrain, judicious use of rebelays and knots are by far the simplest and most foolproof methods (provided you mitigate factor 2 fall risks). If you're trying to push the difficulty while free-climbing, the spring loaded clamp or especially the Gromm hitch systems are very quick to do, but require additional equipment. Friction hitch methods land somewhere in the middle.
