  We consider a two-type reducible branching Brownian motion, defined as a
particle system on the real line in which particles of two types move according
to independent Brownian motions and create offspring at a constant rate.
Particles of type $1$ can give birth to particles of types $1$ and $2$, but
particles of type $2$ only give birth to descendants of type $2$. Under some
specific conditions, Belloum and Mallein in \cite{BeMa21} showed that the
maximum position $M_t$ of all particles alive at time $t$, suitably centered by
a deterministic function $m_t$, converge weakly. In this work, we are
interested in the decay rate of the following upper large deviation
probability, as $t\rightarrow\infty$, \[ {\mathbb P}(M_t\geq \theta m_t),\quad
\theta>1. \] We shall show that the decay rate function exhibits phase
transitions depending on certain relations between $\theta$, the variance of
the underling Brownian motion and the branching rate.
