If a pilot uses an electronic flight bag (say, an iPad with ForeFlight or WingX), are they required to carry paper charts as a backup?
Do different rules apply to operations under Parts 91, 121, and 135?
A EFB is a legal replacement for paper charts and you are not required to carry backup if you are operating as part 91 single engine piston. Advisory Circular 120-76B is in reference to part 91F (Large and Turbine-Powered Multiengine Airplanes).
AC 91-78 is aimed at Part 91 operators, VFR or IFR and states that EFBs can be used in all phases of flight in lieu of paper.
A backup data source is suggested, but is not required. Note that this backup can be another electronic device.
Typically pilots using EFBs (which need to be approved by the FAA if being used for Part 121 or 135 flights) do not need to carry paper charts if, as Steve points out, you have at least two of them. For Part 121, 135, and 91K there is usually a six month period after gaining EFB approval when the FAA requires paper backups in addition to the EFB's as a precaution.
A typical EFB program will have contingency procedures to use in the event of a failure or problem with one or more of the EFBs, and usually says that if one fails (and it can not be replaced or repaired) that you may not depart unless you have paper backups.
Yes, if you only carry one EFB, then you have to also carry paper charts as a backup.
See AC 120-76B.
a. Paper Data Removal. At least two operational EFBs are required to remove paper products that contain aeronautical charts, checklists, or other data required by the operating rules. The design of the EFB function requires that no single failure or common mode error may cause the loss of required aeronautical information.
If paper charts are "required operating information" If you would otherwise be required to carry a paper chart, then you either still have to carry one, or else carry two EFBs.
What about part 91 operators?
Aircraft operated under part 91, except for parts 91F and 91K, require no EFB authorization or compliance with this AC, provided the EFB does not replace any equipment or operating information required by the regulations.