  We consider Boltzmann random triangulations coupled to the Ising model on
their faces, under Dobrushin boundary conditions and at the critical point of
the model. The first part of this paper computes explicitly the partition
function of this model by solving its Tutte's equation, extending a previous
result by Bernardi and Bousquet-M\'elou to the model with Dobrushin boundary
conditions. We show that the perimeter exponent of the model is 7/3 in contrast
to the exponent 5/2 for uniform triangulations. In the second part, we show
that the model has a local limit in distribution when the two components of the
Dobrushin boundary tend to infinity one after the other. The local limit is
constructed explicitly using the peeling process along an Ising interface.
Moreover, we show that the main interface in the local limit touches the
(infinite) boundary almost surely only finitely many times, a behavior opposite
to that of the Bernoulli percolation on uniform maps. Some scaling limits
closely related to the perimeters of finite clusters are also obtained.
