The idiom break with the past is a noun phrase. Yes, "break" has an "action meaning," but the word is still a noun. How do we know that it's a noun? Well, we can turn it plural, adding the plural morpheme -s: breaks with the past.
The idiom can also be represented by a verb phrase, in a different sentence: They are breaking with the past. How do we know that "breaking" is a verb? Well, it has the verb morpheme -ing and it's part of the progressive construction be + ing. In this case, "with the past" is adverbial, referring to the verb.
A separate issue is whether "with the past" (either "adjectival" referring to a noun, or "adverbial" referring to a verb) is a modifier or a complement. That depends on who you ask; not everyone uses the same terms in the same manner. As I said earlier, a simple test used by some is to delete the phrase "with the past." Does:
(a) Making a completely clean break, the couple got rid of all their old furniture
mean the same as
(b) Making a completely clean break with the past, the couple got rid of all their furniture
?
If so, then "with the past" is a modifier. But if you feel that (a) doesn't quite mean the same as (b), that something is missing, then "with the past" is a complement. As a complement, we say that "with the past" is licensed/specified by the noun "break." Since there is no Higher Authority ruling over terminology, people are free to analyze things as they wish. Now, there is a good case for keeping "with the past" in the sentence as a complement. Again, the idiom is "break with the past," and idioms are usually kept intact.