The six known highly dispersed fast radio bursts are attributed to extragalactic radio sources, of unknown origin but extremely energetic. We propose here a new explanation - not requiring an extreme release of energy - involving a body (planet, asteroid, white dwarf) orbiting an extragalactic pulsar. We investigate a theory of radio waves associated to such pulsar-orbiting bodies. We focus our analysis on the waves emitted from the magnetic wake of the body in the pulsar wind. After deriving their properties, we compare them with the observations of various transient radio signals in order to see if they could originate from pulsar-orbiting bodies. The analysis is based on the theory of Alfvén wings: for a body immersed in a pulsar wind, a system of two stationary Alfvén waves is attached to the body, provided that the wind is highly magnetized. When destabilized through plasma instabilities, Alfvén wings can be the locus of strong radio sources convected with the pulsar wind. Assuming a cyclotron maser instability operating in the Alfvén wings, we make predictions about the shape, frequencies and brightness of the resulting radio emissions. Because of the beaming by relativistic aberration, the signal is seen only when the companion is perfectly aligned between its parent pulsar and the observer, as for occultations. For pulsar winds with a high Lorentz factor, the whole duration of the radio event does not exceed a few seconds, and it is composed of one to four peaks lasting a few milliseconds each, detectable up to distances of several Mpc. The Lorimer burst, the three isolated pulses of PSR J1928+15, and the recently detected fast radio bursts are all compatible with our model. According to it, these transient signals should repeat periodically with the companion's orbital period. The search of pulsar-orbiting bodies could be an exploration theme for new- or next-generation radio telescopes.