I have seen electric guitars and basses that don't have any pick ups. For instance, here is an image of two Line 6 electric basses that have no pick ups:
How do these instruments work if they have no pickups?
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As well as the in-bridge piezo pickups (either for acoustic tones or MIDI, which is what the Variax guitars use to feed the modelling system), there is a relatively rare second option: hidden pickups. The "Type I" Fender Marauder, for example, had magnetic pickups concealed under the pick guard:
On the same theme, as of a few days ago, Seymour Duncan are selling a Brad Paisley signature Secret Agent pickup, designed to be concealed under a Fender Esquire pickguard, expanding it from one pickup to two.