Locating the main verb of a subordinate that-clause

Question

In this CNN report starting at 2:58, a British reporter says:

Trump is so deeply unpopular that her sort of gilded way of smiling in his presence, never really causing a standoff, and managing to perhaps keep the whole thing afloat. The special relationship is vital to so many Britons here frankly.

If the above transcript is correct, what is the main verb of the that-clause?

...that her sort of gilded way of smiling in his presence, never really causing a standoff, and managing to perhaps keep the whole thing afloat.

If there's anything wrong with the transcript, please let me know.

EDIT

I don't understand what @JeremyC means "there is no verb in the that clause". There ARE at least four verbs in the that-clause: smiling, causing, managing, and keep.

The question is whether any one of these verbs is acting as a main verb. Clearly, smiling is not, because the VP smiling in his presence is a complement of the preposition of. Neither is keep, because the VP to perhaps keep the whole thing afloat is a complement of the verb managing.

Then, the question boils down to whether the verbs causing and managing are acting as the main verbs of the that-clause, or whether the two verbs are also complements of the preposition of as well.

So, which is it going to be?

Answer 1

Your transcription is incomplete.  Although mine's an American ear unaccustomed to this accent, the sentence I hear is this:

Trump is so deeply unpopular that her sort of gilded way of smiling in his presence, never really causing a stand-off and managing to perhaps keep the whole thing afloat -- this special relationship is vital to so many Britons here, frankly -- may mean she's the ultimate winner for me.

The predicative construction in the clause in question is "may mean".  The finite verb here is the modal auxiliary "may". 

The subject of this clause is quite long and involved: "her sort of gilded way of smiling in his presence, never really causing a standoff, and managing to perhaps keep the whole thing afloat".  This subject does contain a number of verbs, in the sense of that part of speech.  "Gilded" is a participle.  "Smiling", "causing", and "managing" are coordinate gerunds.  "To ... keep" is an infinitive.  None of these are finite forms.  In the sense of the parts of a sentence, none of those are verbs. 

There happens to be a parenthetical independent clause between this long and involved subject and the verb with which it agrees.  The aside "this special relationship is vital to so many Britons here, frankly" is merely an interruption.  It is not enough to end the clause in question.  When the interruption itself ends, the verb of the interrupted clause begins. 

Had your transcription been correct, that would have indicated that the reporter lost his train of thought before he finished his sentence.  We'd have been left with a subject stranded from its forgotten predicate.  That didn't happen here.  The sentence is complete.  It simply extends past the end of the transcript you provided. 

Answer 2

The sentence is a good example of the well-known phenomenon that even highly educated native speakers of a language do not always speak in well-formed grammatical sentences. It also well-known that native-speakers of languages tend not to notice such "errors" when they hear them.

But when the speech is transcribed so the spoken word becomes written, then we see that something has gone "wrong".

In this case, as you have noticed, there is no verb in the that clause. Does it matter? Maybe not. Ungrammatical as the sentence is, the reporter has nevertheless conveyed his thought that the person described (Mrs May, perhaps?) has found an effective way of managing her relationship with Mr Trump.