  In this text, we study Viro's conjecture and related problems for real
algebraic surfaces in $(\mathbb{CP}^1)^3$. We construct a counter-example to
Viro's conjecture in tridegree $(4,4,2)$ and a family of real algebraic
surfaces of tridegree $(2k,2l,2)$ in $(\mathbb{CP}^1)^3$ with asymptotically
maximal first Betti number of the real part. To perform such constructions, we
consider double covers of blow-ups of $(\mathbb{CP}^1)^2$ and we glue singular
curves with special position of the singularities adapting the proof of
Shustin's theorem for gluing singular hypersurfaces.
