  Twisting two monolayers of a high-$T_c$ cuprate superconductor can engender a
chiral topological state with spontaneously broken time reversal symmetry
$\mathcal{T}$. A crucial ingredient required for the emergence of a gapped
topological phase is electron tunneling between the CuO$_2$ planes, whose
explicit form (in an ideal clean sample) is dictated by the symmetry of the
atomic orbitals. However, a large body of work on the interlayer transport in
cuprates indicates importance of disorder-mediated incoherent tunneling which
evades the symmetry constraints present in an idealized crystal. The latter
arises even in the cleanest single-crystal samples through oxygen vacancies in
layers separating the CuO$_2$ planes, introduced to achieve the hole doping
necessary for superconductivity. Here we assess the influence of incoherent
tunneling on the phase diagram of a twisted bilayer. We show that the model
continues to support a fully gapped topological phase with broken
$\mathcal{T}$, even in the limit of disorder-mediated interlayer coupling.
Compared to the model with a constant, momentum conserving interlayer coupling,
the extent of the topological phase around the 45$^\circ$ twist decreases with
increasing incoherence, but remains robustly present for parameters likely
relevant to Bi$_2$Sr$_2$CaCu$_2$O$_{8+\delta}$.
