  A branching random tessellation (BRT) is a stochastic process that transforms
a coarse initial tessellation of $\mathbb{R}^d$ into a finer tessellation by
means of random cell divisions in continuous time. This concept generalises the
so-called STIT tessellations, for which all cells split up independently of
each other. Here, we allow the cells to interact, in that the division rule for
each cell may depend on the structure of the surrounding tessellation.
Moreover, we consider coloured tessellations, for which each cell is marked
with an internal property, called its colour. Under a suitable condition, the
cell interaction of a BRT can be specified by a measure kernel, the so-called
division kernel, that determines the division rules of all cells and gives rise
to a Gibbsian characterisation of BRTs. For translation invariant BRTs, we
introduce an "inner" entropy density relative to a STIT tessellation. Together
with an inner energy density for a given "moderate" division kernel, this leads
to a variational principle for BRTs with this prescribed kernel, and further to
an existence result for such BRTs.
