Comic Blunt Card with how used as a complementizer, roughly like that:
(Hat tip to Chris Ambidge.)
There is now some literature on complementizer how, in particular
Legate, Julie Anne. 2010. On how how is used instead of that. NLLT 28.1.121-34.
with references to (the few) earlier mentions, in particular to the treatment of it in the Cambridge Grammar of the English Language, in Rodney Huddleston’s chapter on “Content clauses and reported speech”:
In very informal style how can be used without any trace of its usual manner (or degree) meaning, and in such cases it is arguable that it is no longer an interrogative word but has been reanalysed as a declarative subordinator, a variant of that … how here is simply equivalent to that (p. 954)
Now, readers of my many pieces on “unfree variation” will be able to guess that I’m going to be suspicious of the claim of equivalence with only stylistic differentiation, citing:
Bolinger’s Dictum [original version]: A difference in syntactic form always spells a difference in meaning.
Bolinger’s Dictum [AMZ’s hedged version]: Lexical and syntactic variation is unfree; variants usually have (subtly) different meanings or discourse functions, which can be observed in certain contexts (though these differences might not be of consequence in many contexts).
In fact, I observe that in many cases claimed free variants turn out to be syntactically distinguished as well. So it is in this case. As Legate puts it,
the how-clause exhibits systematically distinct behaviour not only from the corresponding embedded CP headed by that (henceforth that-clause), but also from embedded interrogative CPs. Specifically, the how-clause behaves as a definite DP [NP in older syntactic literature and in CGEL] with presupposed propositional content. (p. 122)
In particular, how-clauses can be objects of prepositions, while that-clauses cannot:
They told me about ✓how / *that the tooth fairy doesn’t really exist.
Legate also notes that the way that usually provides a better paraphrase for complementizer how than plain that does — as in the cartoon above:
I like the way that when you call me ‘bro’ when we’re drunk, it makes what we’re doing seem less gay.
More details in Legate’s article.
[There's another feature of linguistic interest in the cartoon sentence: summative it, referring to the other man's calling the speaker 'bro' when they're both drunk, so that the sentence can be roughly paraphrased as
I like how your calling me 'bro' when we're drunk makes what we're doing seem less gay
But only roughly, since the briefer version lacks the explicit topic-introducing function of when you call me 'bro'.]