Do Bayesian Network perfect maps need to be chordal? In Probabilistic Graphical Models by Koller and Friedman, there is a proposition -

The PDAG $\mathcal K$ returned by Build-PDAG is necessarily chordal.

Build-PDAG is an algorithm that builds the equivalence-class-PDAG for the perfect map, if one exists.
My confusion is with the definition of chordal graph.
If a PDAG is chordal, when its undirected version is chordal (which is the definition the text also gives), it implies that a perfect map of below type is not possible. Whence my question - do BN perfect maps need to be chordal.

However, if chordal graph is only for undirected graphs, then the above theorem is appropriate. The loop $A-B-X-C-A$ will definitely contain an immorality and hence the edges for the immorality will become oriented by Build-PDAG.

No. Bayesian Network perfect maps need not be chordal. And your example is a case in point. At least, if chordality is really defined as the chordality of the undirected version.
So this is a bit confusing. But in the text following the cited statement, what they actually use is the fact that the undirected components of the chain graph $\mathcal K$ are chordal. So my guess is that that's what they refer to by "$\mathcal K$ is chordal".
Then, in your example, $\mathcal K$ would be the chain graph:

and the only nontrivial undirected component present is:

And this component is indeed chordal. And that, IIUC, is all they need in their proof of Theorem 3.10.
