Learn English – Deaths of both Romeo and Juliet or death of both Romeo and Juliet

There are several types of phrasal verb, and several types of verb + preposition. Not all of them are constituents, they serve different functions, they're all irregular as hell, and they're all governed by the matrix verb.

Every verb has its own assortment of special idioms, affordances, prohibitions, requirements, and irregularities. There is such immense variation in these details that such matters are considered part of the meaning of the verb; certainly they are strongly affected by the meanings. Square and cube that variation for phrasal verbs, since there are far more phrasal verbs in English than there are single-word verbs.

Some varieties can be examined in this freshman grammar homework problem. Examples of different types, from there:

Sentences (1) and (2) show two normal verb + prep constructions, from the same verb: look at, with transitivizing at; and look for, a transitive idiom meaning 'search'. Both of them require that the preposition precede the object (which may be thought of either as the object of the preposition, or as the direct object of the transitive verb + prep construction), even if that object is a pronoun. It makes no difference to most prepositions whether their object is a noun or a pronoun (ungrammatical sentences are marked with an asterisk *):

• I looked for Einstein ~ *I looked Einstein for ~ I looked for him ~ *I looked him for.

Sentence (3), on the other hand, is a real transitive phrasal verb. There are two characteristics of phrasal verbs that help to distinguish them. Both tests have limitations, however. The most important one, and the easiest test to administer, is the difference between the pattern of asterisks in the second and third columns, where pronoun objects force the difference.

There is a syntactic rule (called Particle Shift in the literature) that applies to transitive phrasal verbs only, and imposes a special requirement on pronoun objects. Thus, with a real phrasal verb like look up 'research (v)', the particle may appear either before or after a Noun object, but must appear after a Pronoun object.

• I looked up Einstein ~ I looked Einstein up ~ *I looked up him ~ I looked him up.

Note, however, that this test is helpful only with transitive phrasal verbs. There are plenty of intransitive phrasal verbs, too, but there's no object to test with. Many transitive phrasal verbs can appear also intransitively, e.g take off, move away, often with a different sense (He took it off ~ The plane took off), or not (He moved it away ~ It moved away).

The second useful characteristic is that a phrasal verb is stressed on the particle, at least as much as on the verb, and maybe more. A V + PP construction like look at, on the other hand, is stressed on the verb, not the preposition.

• He looked up the word. ~ He looked at the word.

That's because prepositions are rarely stressed, except for emphasis (In the toilet, you idiot!);
they're sposta slide by like articles and conjunctions to grease the way into the object, which is the informational part. They're not sposta distract, so they're unstressed, and therefore reduced, so we get common contractions like sposta and lookit.

Unfortunately, stress is not represented in English writing, so that distinction is not helpful for readers.

The thing is the books.

As the preponderance of comments suggests, an appropriate way to describe the grammar of this sentence, along with a host of others just like it, is: it is aces:

Exclamation that expresses something as being exceedingly good.
Urbandictionary.com

Divorcing the reduced construction from its larger context might give some native speakers pause, because the reduction eliminates contextual signals of the idiomatic usage:

thing

noun

6 (the thing) informal Used to introduce or emphasize an important point:

ODO

One might argue that the thing is the books is inappropriate for a formal lecture or paper, but one cannot argue that it is grammatically defective. The reduced expression is correct, and in its original context, it doesn't sound the least bit strange:

The only thing that I want you to hit right now is the books.

One might forgive an untrained writer for the erroneous proximity agreement, and even decipher the meaning of it in the full sentence:

*The only thing that I want you to hit right now ?are? the books.

But it is, in fact, erroneous proximity agreement between the verb and the predicative complement, as the reduced construction reveals:

*The thing ?are? the books.

It is one thing to forgive an error. It is quite another thing to teach an error, and the OP's rule may seem correct under cursory inspection, but it is spurious. Must the subject and subject complement match in number?

The only reason the thing is the books might sound strange is that our ears are more familiar with common predicative expressions of similar construction:

• The girls are musicians. But not: The girls are musician.
• The car is a clunker. But not: The car is [the] clunkers.
• The actors are men. But not: The actors are man.

Even if the vast majority of predicative complements happens to match the number of the subject, there is no universal syntactical rule requiring it. Consider how simple it is to generate exceptions to this imaginary rule:

• Taxes are the issue. Or The issue is taxes.
• The prices are my concern. Or My concern is the prices.
• The mosquitos are his problem. Or His problem is the mosquitos.
• Her clothes are the point. Or The point is her clothes.
• Flowers are today's subject. Or Today's subject is flowers.

In the five constructions above, mismatching the number of the subject and predicative complement works in both directions:

plural X are singular Y
AND
singular Y is plural X.

The reason is quite simple: regardless of which is the subject, the predicative syntax cooperates with the semantics of the singular noun to put the plural noun in a single collective bucket. An idiomatic usage of the thing exerts the same collectivizing influence on the books:

• The books are the thing. Or The thing is the books.

The thing is singular, and the writer has chosen to equate the books with the thing. Apparently, the writer intends the reader to recognize the collective nature that turns the books into the thing. Should the reader respect the intentions of the writer, or cast pseudo-grammatical insults at the writer?

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Prescriptive grammarians presume to keep our language "manageable" by giving arbitrary orders to native speakers. Deep in their intuitive understanding of how English really works, native speakers know better, but tragically, they cave in to the intimidation of seventy-five-cent obfuscations like morphosyntactic plurality conflict. [People who can say such big words with presumptive authority probably know more than us, right?]

Since there is no morphosyntactic plurality conflict in the sentence, what is it that bothers the prescriptive grammarian? It is the words, the intelligent arrangement of the words, and the intelligent speaker's refusal to fill the erroneous prescription. Please notice the previous sentence:

It is the words [that bother the prescriptive grammarian].

We might call it a dummy-subject, but it is singular, because the larger context invited it to match the interrogative what is it. The syntax of English requires the verb to match the singular subject with the singular form: is. After that, the predicative complement offers an intelligent answer to the question: the words bother the prescriptive grammarian. Again, notice how the larger context welcomes the plural words, because a long list of perfectly intelligent constructions irritates prescriptive grammarians.

The coup de grâce for the prescriptive grammarian's imaginary morphosyntactic plurality conflict is that the sentence answers the question with a compound complement in addition to a plural complement:

It is the words, the intelligent arrangement of the words, and the intelligent speaker's refusal to fill the erroneous prescription.

Semantics is about the meaning of words. Syntax is about the intelligent arrangement of words in sentences. When native speakers understand they are not sick, they just refuse to fill the erroneous prescription. The thing is the books is not at all grammatically defective, and the prescriptive grammarian exposes his own ignorance of English grammar by asserting the imaginary defect.

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Question 1: If we explain this phenomena [sic] in terms of notional vs syntactic agreement, where does "grammatically correct/incorrect" fit in?

The premise of contradiction is flawed. From grammar.about.com, the definition of notional agreement is:

Agreement (or concord) of verbs with their subjects and of pronouns with their antecedent nouns on the basis of meaning rather than grammatical form. Also known as synesis.

The singular verb is agrees with the subject, so it is agreement, not "notional agreement." Even if there were a universal syntactical rule requiring the agreement of the subject and the predicative complement, it would be a discrete rule with no hard and fast bearing on the agreement between subject and verb. Regardless, the predicative syntax operates with the semantics of the singular subject to collectivize the plural predicative complement.

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Question 2: Can a simple sentence such as "The thing is the objects." be grammatically incorrect while more complex sentences such as "The thing is four pieces of driftwood glued together." be grammatically correct? (The answer, in my mind, must be no; this is my conundrum.)

Although confused prescriptive grammarians might hope to salvage their sullied reputations, the simple sentence is just as correct as the more complex sentence, because in both situations, the predicative syntax operates with the semantics of the singular noun to collectivize the plural noun.

In the more complex sentence, the the semantics and syntax of the modifying phrase glued together reinforce the collectivization of the perfectly grammatical reduced expression: The thing is four pieces of driftwood. Please, notice that even this non-idiomatic use of The thing works quite well to break the imaginary rule about matching the number of subject and complement.

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Question 3: Would it be better to say that Sentence 2.a has a low level of (linguistic) grammaticality?

Since the expression is perfectly grammatical, it might be better to say that the prescriptive grammarian making such a claim has a low level of insight into English grammar.

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Question 4: How about a low level of gradient well-formedness and that it's semantically difficult to be (linguistically) acceptable?

Since the expression is well-formed and quite easy to understand, only a frustrated prescriptive grammarian refusing to abandon an imaginary rule would find it difficult to accept.

What is truly unacceptable is instructing students to break a real syntactic rule for the sake of applying a spurious rule. It would seem acceptable to lay out a descriptive rule for students: The subject complement normally agrees with its subject.

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Question 5: Do we simply draw a hard line and say "The thing is the objects." is 100% grammatically correct? I didn't particularly like this option based on intuitive notions of "grammatically correct".

Intuition informed by error is erroneous intuition. Native English-speaking Everyman has drawn an intelligent intelligible line, and most intelligent people are delighted to talk and write on Everyman's side of the line. What is it that bothers the prescriptive grammarian?

It is semantics, syntax, and the intelligent speaker's refusal to fill an erroneous prescription.