  In this work we study the bisexual Galton-Watson process with a finite number
of types, where females and males mate according to a ''mating function'' and
form couples of different types. We assume that this function is superadditive,
which in simple words implies that two groups of females and males will form a
larger number of couples together rather than separate. In the absence of a
linear reproduction operator which is the key to understand the behaviour of
the model in the asexual case, we build a concave reproduction operator and use
a concave Perron-Frobenius theory to ensure the existence of eigenelements.
Using this tool, we find a necessary and sufficient condition for almost sure
extinction as well as a law of large numbers. Finally, we study the almost sure
long-time convergence of the rescaled process through the identification of a
supermartingale, and we give sufficient conditions to ensure a convergence in
$L^1$ to a non-degenerate limit.
