Stefan Monnier writes:

>> Investigating how to fix this leads to the longer story.  There are two
>> possible approaches:
>>
>>   1) use a string-style syntax (generic string) instead of c-style
>>      comments to flag HERE-documents.  That way, font-lock picks up the
>>      correct face automagically.
>>
>>   2) Keep HERE_docs as c-style comments, but change the face mapping by
>>      injecting a function into font-lock-defaults which applies the
>>      string face to c-style comments.
>
> 3) Use `font-lock-syntactic-face-function`?

Ah - thanks for this pointer!  I wasn't aware of this function, though
this feature is already in use in perl-mode.el.

This looks like to be an improved variation of 2): HERE-docs remain
marked as c-style comments, and `font-lock-syntactic-face-function` is
used to display them as strings.

A patch for this variation is attached.  Tests and test resources are
the same as with the first patch, and this patch also contains the same
fix for HERE-doc starters with a trailing comment.
-- 
Cheers,
haj