  Using toric geometry we give an explicit construction of the compact steady
solitons for pluriclosed flow first constructed in arXiv:1802.00170. This
construction also reveals that these solitons are generalized K\"ahler in two
distinct ways, with vanishing and nonvanishing Poisson structure. This gives
the first examples of generalized K\"ahler structures with nonvanishing Poisson
structure on non-standard Hopf surfaces, completing the existence question for
such structures. Moreover this gives a complete answer to the existence
question for generalized K\"ahler-Ricci solitons on compact complex surfaces.
In the setting of generalized K\"ahler geometry with vanishing Poisson
structure, we show that these solitons are unique. We show that these solitons
are global attractors for the generalized K\"ahler-Ricci flow among metrics
with maximal symmetry.
