  In this paper we define the notion of a generalized coK\"ahler structure and
prove that the product $M_{1}\times M_{2}$ of generalized contact metric
manifolds $(M_i, \Phi_i,E_{\pm,i}, G_i)$, $ i=1, 2$, where $M_{1}\times M_{2}$
is endowed with the product generalized complex structure induced from $\Phi_1$
and $\Phi_2$, is generalized K\"ahler if and only if $(M_i, \Phi_i, E_{\pm,i},
G_i) ,\ \ i=1,2$ are generalized coK\"ahler structures. We also prove that
products of generalized coK\"ahler and generalized K\"ahler manifolds admit a
generalized coK\"ahler structure. We use these product constructions to give
nontrivial examples of generalized coK\"ahler structures. Finally, we show the
analogs of these theorems hold in the setting of twisted generalized
geometries. We use these theorems to construct new examples of twisted
generalized K\"ahler structures on manifolds that do not admit a classical
K\"ahler structure and we give examples of twisted generalized coK\"ahler
structures on manifolds which do not admit a classical coK\"ahler structure.
