  We discuss the phase diagram of the extended Hubbard model with both
attractive and repulsive local and nonlocal interactions. The extended
dynamical mean-field theory (EDMFT) and the dual boson method (DB) are
compared. The latter contains additional nonlocal correlation effects that are
not incorporated in EDMFT. We find that EDMFT and DB give almost identical
results in the attractive $V$ regime, where phase separation occurs. This is
quite a difference with the previously studied repulsive $V$ regime, where
EDMFT and DB give very different phase boundaries for the checkerboard order
phase, especially at small $U$.
