This is ambiguous in writing but completely unambiguous in speech. In the example sentence that is a noun phrase, not a subordinator. It is easier to see how the sentence works if we change the word that. This word talks about something in the previous sentence. Let's use the word this, which works the same way, instead:
- We saw this [when Chinese warships turned up off the coast of Alaska within U.S. territorial waters, at the same time that President Obama was on the ground here].
This sentence is a transcript of a spoken report by a journalist for CNN. When we read the sentence it is easy to think that the word that is a subordinator. Compare the following sentences:
- I saw that yesterday.
- I saw that she had already left.
In the first sentence that means that thing. In the second sentence the word that introduces a new clause. Notice that in the second sentence the verb SEE changes its meaning; it means something like NOTICE or UNDERSTAND.
In the Original Poster's example, the word that means that thing, it doesn't introduce a new clause. Now, when we read the sentence, the word that is ambiguous until we get to the end of the sentence and then realise that it cannot be the subordinator. However, this sentence is from a spoken report. In the actual speech, this sentence will not be ambiguous.
The reason it won't be ambiguous is that the noun phrase that is always stressed and has a full vowel, the same vowel that we find in the word cat. The IPA looks like this: /ðæt /. The subordinator that is almost never stressed and is almost always pronounced /ðət/ with the same vowel that we find at the beginning of the word America. We almost never confuse these two words in speech.