  The $2$d orders are a sub class of causal sets, which is especially amenable
to computer simulations. Past work has shown that the $2$d orders have a first
order phase transition between a random and a crystalline phase. When coupling
the $2$d orders to the Ising model, this phase transition coincides with the
transition of the Ising model. The coupled system also shows a new phase, at
negative $\beta$, where the Ising model induces the geometric transition. In
this article we examine the phase transitions of the coupled system, to
determine their order, as well as how they scale when the system size is
changed. We find that the transition at positive $\beta$ seems to be of mixed
order, while the two transitions at negative $\beta$ appear continous/ first
order for the Ising model/ the geometry respectively. The scaling of the
observables with the system size on the other hand is fairly simple, and does,
where applicable, agree with that found for the pure $2$d orders. We find that
the location of these transitions has fractional scaling in the system size.
