  Unlike in hyperbolic geometry, the monodromy ideal triangulation of a
hyperbolic once-punctured torus bundle $M_f$ has no natural geometric
realisation in Cauchy-Riemann (CR) space. By introducing a new type of
$3$--cell, we construct a different cell decomposition $\mathcal{D}_f$ of $M_f$
that is always realisable in CR space. As a consequence, we show that every
hyperbolic once-punctured torus bundle admits a branched CR structure, whose
branch locus is the set of edges of $\mathcal{D}_f$. Furthermore, we explicitly
compute the ramification order around each component of the branch locus and
analyse the corresponding holonomy representations.
