First, I’d advise you to drop the term 'phrasal verb'. It’s thoroughly misleading. It’s not the whole expression "call for" that is a verb; it’s just the lexeme "call".
Your example, This calls for a thorough examination of the facts is analysed as verb - [prep + O]. It’s a verbal idiom headed by the prepositional verb “call”, where the preposition “for” is specified by the verb.
Particles are intransitive prepositions (plus a few adjectives and verbs) which can freely come between a transitive verb and its direct object. The distinctive property of particles is that they can precede the object but in general they don’t have to, so there is alternation between, say, Ed took off his coat and Ed took his coat off. Now compare that pair with your example This calls for an examination” ~ * This calls an examination for. Clearly "for" is not a particle on two counts: in that example, "for" is a transitive preposition and "call" is an intransitive verb.
Complements vs adjuncts:
Complements in NP structure are virtually restricted to PPs (usually headed by "of") and subordinate clauses. In your example, the PP "of the facts" is complement of the noun "examination". In NPs, dependents with the form of PPs qualify as complements when they are licensed by the particular head noun. In your example, the PP "of the facts" is licensed by "examination". The clearest of cases have one or more of these properties:
(1) The PP corresponds to object or subject NPs, compare the facts examined ~ the examination of the facts.
(2) The choice of preposition is specified by the head noun, as is the case with your example "call for a thorough examination", where "call" specifies "for".
(3) the PP is obligatory because the noun makes little sense without it: "the advent of the steam engine", "the feasibility of the proposal", "a dearth of new ideas" and so on.
Complements are a kind of dependent that must be licensed by the head of the phrase. By contrast, adjuncts require no such licensing; they function as modifiers in clause structure, or as supplements. Obligatory elements are always complements: they are needed to complete the verb phrase; optional elements may be either complements or adjuncts.
Matthews in The Concise Oxford Dictionary of Linguistics defines complement as “A syntactic element seen as completing the construction of another element”: the syntactic concept construction is preferable to the semantic concept meaning. He goes on to say, more specifically, that it applies to elements which are within the valency of a verb or other lexical unit. And his entry for "valency" is: “The range of syntactic elements either required or specifically permitted by a verb or other lexical unit"– which is equivalent to licensing.