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We consider a Galton-Watson process $\mathbf{Z}% (n)=(Z_{1}(n),Z_{2}(n))$with two types of particles. Particles of type 2 may produce offspring of bothtypes while particles of type 1 may produce particles of their own type only.Let $Z_{i}(m,n)$ be the number of particles of type $i$ at time $m<n$ havingoffspring at time $n$. Assuming that the process is critical and that thevariance of the offspring distribution may be infinite we describe theasymptotic behavior, as $m,n\rightarrow \infty $ of the law of$\mathbf{Z}(m,n)=(Z_1(m,n),Z_2(m,n))$ given $\mathbf{Z}(n)\neq \mathbf{0}$. Wefind three different types of coexistence of particles of both types. Besides,we describe, in the three cases, the distributions of the birth time and thetype of the most recent common ancestor of individuals alive at time$n\rightarrow \infty .$

Reduced two-type decomposable critical branching processes with possibly infinite variance