  In this paper, we develop the notion of free-Boolean independence in an
amalgamation setting. We construct free-Boolean cumulants and show that the
vanishing of mixed free-Boolean cumulants is equivalent to our free-Boolean
independence with amalgamation. We also provide a characterization of
free-Boolean independence by conditions in terms of mixed moments. In addition,
we study free-Boolean independence over a $C^*$-algebra and prove a positivity
property.
