  We study the separable complementation property (SCP) and its natural
variations in Banach spaces of continuous functions over compacta $K_{\mathcal
A}$ induced by almost disjoint families ${\mathcal A}$ of countable subsets of
uncountable sets. For these spaces, we prove among others that $C(K_{\mathcal
A})$ has the controlled variant of the separable complementation property if
and only if $C(K_{\mathcal A})$ is Lindel\"of in the weak topology if and only
if $K_{\mathcal A}$ is monolithic. We give an example of ${\mathcal A}$ for
which $C(K_{\mathcal A})$ has the SCP, while $K_{\mathcal A}$ is not monolithic
and an example of a space $C(K_{\mathcal A})$ with controlled and continuous
SCP which has neither a projectional skeleton nor a projectional resolution of
the identity. Finally, we describe the structure of almost disjoint families of
cardinality $\omega_1$ which induce monolithic spaces of the form $K_{\mathcal
A}$: They can be obtained from countably many ladder systems and pairwise
disjoint families applying simple operations.
