A binary planet system consisting of two terrestrial planets (hereafter Alpha/α and Beta/β). The system barycentre (C) is between the two planets; for the sake of this question the exact position is variable as long as it remains in the space between the two, but closer to Alpha than to Beta. They are tidally locked to each other.
Alpha is a continental planet comparable in all aspects to Earth (i.e. within 20% of size, mass, surface gravity, etc), with tectonic activity at least as active as on Earth (e.g. global average tectonic plate speed is no less than ~7 cm/year).
Ideally Beta is similar, but consider its dimensions and features as variable (so long as it remains a terrestrial planet) as needed.
Which – if any – of these scenarios is the most plausible?
A dense ocean plate becomes fixed below the barycentre in its area of influence. The plate may drift or rotate within this area.
A thick and mountainous continental plate becomes fixed below the barycentre in its area of influence. The plate and its mountains may drift or rotate within this area.
A hot spot or upwelling from the mantle becomes a fixed feature below the barycentre in its area of influence. Plates drift freely, causing a chain of volcanoes. The oldest volcano attributed to this hot spot is no less than 100 million years old.
Before you say none of the above...
(And I'm completely open to that answer.) Would positioning the barycentre directly on the surface of Alpha make a difference?