  This paper studies the interplay between self-crossing boundary Lefschetz
fibrations and generalized complex structures. We show that these fibrations
arise from the moment maps in semi-toric geometry and use them to construct
self-crossing stable generalized complex four-manifolds using Gompf--Thurston
methods for Lie algebroids. These results bring forth further structure on
several previously known examples of generalized complex manifolds. We moreover
show that these fibrations are compatible with taking connected sums, and use
this to prove a singularity trade result between two types of singularities
occurring in these fibrations.
