  We propose a topological qubit in which braiding and readout are mediated by
the $4\pi$ Majorana-Josephson effect. The braidonium device consists of three
Majorana nanowires that come together to make a tri-junction; in order to
control the superconducting phase differences at the tri-junction the nanowires
are enclosed in a ring made of a conventional superconductor; and in order to
perform initialization/readout one of the nanowires is coupled to a fluxonium
qubit through a topological Josephson junction. We analyze how flux-based
control and readout protocols can be used to demonstrate braiding and qubit
operation for realistic materials and circuit parameters.
