  In this paper, we introduce the notion of conditionally bi-free independence
in an amalgamated setting. We define operator-valued conditionally
bi-multiplicative pairs of functions and construct operator-valued
conditionally bi-free moment and cumulant functions. It is demonstrated that
conditionally bi-free independence with amalgamation is equivalent to the
vanishing of mixed operator-valued bi-free and conditionally bi-free cumulants.
Furthermore, an operator-valued conditionally bi-free partial
$\mathcal{R}$-transform is constructed and various operator-valued
conditionally bi-free limit theorems are studied.
