This sentence has three important factors to consider: it is an existential construction, the displaced subject of the sentence is a noun phrase coordination, and the coordination is a list (more than two items).
In this particular case, is would be the safest choice (i.e., least likely to run afoul of any rules), although The Cambridge Grammar of the English Language suggests that are would also be acceptable, due to it being a list.
TL;DR:
CGEL says either is or are is acceptable for your sentence.
Collins COBUILD states explicitly that is should be used.
From the Cambridge Grammar of the English Language (2002):
Typically the subject and verb should agree (pg 499).
There are however "semantically motivated overrides." One such override is when the subject is a coordination ("two or more elements of syntactically equal status... linked by means of a coordinator such as and or or"):
Ch 5, §18.4(a) Coordination with and
In general a subject with the form of a coordination of [noun phrases] linked by and takes a plural verb, as in Mary and John are here, etc. It doesn't matter whether the individual coordinates are singular or plural: the coordination as a whole here denotes a set containing at least two members, and hence takes a plural verb.
There are times though when "the subject is conceptualized as a single unit and this determines the singular verb." One of the examples given is, "Eggs and bacon is my favorite breakfast." In that particular example, "my favorite breakfast can only apply to eggs and bacon as a unit and hence the plural verb is impossible." (see also Ch15 §1.3.2 "Joint coordination", which discusses distributive vs. joint coordination in more depth)
The closest noun is however taken into account sometimes when using or (Ch 5 §18.4(b)) and always taken into account with a negative (Ch5 §18.4(d) and (e)).
The example you've given in the question though, is an existential construction (there is a "dummy pronoun" for the "displaced subject": rice, meat and tomatoes). (pg 241, Ch 4 §3.2.2)
Usually the verb should agree with the displaced subject (see footnote 71 on page 500):
There are tomatoes on my plate.
However, when the displaced subject is a coordination, there is an exception (footnote 21 on page 242). When the coordination consists of only two items (binary coordination) then it is idiomatic for the verb to agree with the first item in the list:
A further complication arises in existentials when the verb is followed by an NP-coordination, as in There was/?were a bottle of wine and several glasses on the table. Were tends to be unidiomatic with an NP-coordination when the coordinate that is adjacent to it is singular, even though the coordinate as a whole (a bottle of wine and several glasses) is plural.
A plural example is given on page 1393 [7i]:
There are good teachers and bad teachers.
The exception to the exception is when the coordination consists of a list (three or more items, aka multiple coordination). The last caveat in footnote 21 states that in such a case,
Plural agreement, however, occurs readily in lists: There are still Brown, Jones, Mason and Smith to interview.
Araucaria found an example on page 1400 demonstrating that a singular verb is also valid with lists (the use of "readily" in the footnote 21 is ambiguous about which is more common or preferred), [29i]:
A: Who was at the party last night? B: There was Mary, Sue, Fred, Matt and Sam.
So, with three items in your displaced subject (rice, meat and tomatoes), you can really choose either is or are (according to CGEL).
If there were only two items in the coordination, then it would be less ambiguous; a singular verb would be preferred/idiomatic. Also, note that a plural verb would not necessarily be ungrammatical, indicated by an asterisk, but merely questionable/unidiomatic, indicated by a question mark (see page xii for those notational conventions).
More information can also be found in Chapter 16 "Information packaging", section 6 "Existential and presentational clauses" on page 1390.
Collins COBUILD English Grammar states that with lists the verb should agree with the first item in the list.
10.50... You use a singular form of 'be' when you are giving a list of items and the first noun in the list is singular or uncountable.
1In my humble opinion, I believe that this answer overlooks footnote 21 (Ch 4, §3.2.2, page 242). In the text, "As far as the subject-verb agreement rule is concerned, therefore, it is there which counts as subject: the complication is that it inherits its agreement features from the NP it displaces as subject.[21]" – Damkerng T. – 2015-09-17T12:11:48.423
1(cont.) Footnote 21: "A further complication arises in existentials when the verb is followed by an NP-coordination, as in There was/were a bottle of wine and several glasses on the table. Were tends to be unidiomatic with an NP-coordination when the coordinate that is adjacent to it is singular, even though the coordinate as a whole (a bottle of wine and several glasses) is plural. Plural agreement, however, occurs readily in lists: There are still Brown, Jones, Mason and Smith to interview." – Damkerng T. – 2015-09-17T12:45:40.570
@DamkerngT. thank you for pointing that out; I did miss that. But, I think my answer is still correct, because even in the footnote you mention, it says, "Plural agreement, however, occurs readily in lists: 'There are still Brown, Jones, Mason and Smith to interview.'" and "rice, meat and tomatoes" is a list. – John B – 2015-09-17T15:14:25.147
3@DamkerngT. It’s curious that no native speaker would ever say “There ∗is just three things you need to know”, and yet we often enough use there is in other plural situations: “There’s a man and a woman here to see you.” – tchrist – 2015-09-17T18:49:38.007
1And would you really say "there are an egg and a tomato on my plate"? I'm betting 90%+ would go for "there is an egg and a tomato on my plate", even though those aren't one unit (substitute 'egg' for 'hammer'). – abligh – 2015-09-17T19:29:14.577
1I read the same paragraph and I don't think that CGEL suggests the same suggestion you suggest: "So the verb should agree with the displaced subject". Note that you omitted crucial part in that paragraph: "..., [we call respectively the extraposed subject and the displaced subject.] These terms are intended to capture the fact that they are semantically like the subject of their basic counterpart, but that are not to be interpreted as kinds of subject." In short, I think CGEL would say that "rice, tomatoes and meat" in the OP's sentence is like an imitation diamond. – Damkerng T. – 2015-09-17T19:47:39.307
@DamkerngT. Ah yes I think you're right. After re-reading, I agree with you that the subject is definitely the existential subject, there. However, please see the footnote on page 500 "In a similar way, the dummy pronoun there, takes on the agreement properties of the displaced subject."
– John B – 2015-09-17T20:10:25.7801Thank you for the pointer (page 500)! Footnote 71 indeed supports your suggestion, but wouldn't this get us back to where we started, footnote 21? -- To be honest, I accept both alternatives (as I stated in my old answer), in slight favor of is in the OP's sentence. – Damkerng T. – 2015-09-17T20:22:49.623
@DamkerngT. But even in footnote 21, it says "Plural agreement, however, occurs readily in lists: There are still Brown, Jones, Mason and Smith to interview." – John B – 2015-09-17T20:32:38.597
1That's the very point why I don't strongly object to your answer. I haven't read CGEL before, so I'm not quite sure about the definition of an "NP-coordination" and a "list". It looks (to me) like that besides being a list, Brown, Jones, Mason and Smith (and thus, rice, meat and potatoes/tomatoes) could or should be considered an NP-coordination too? Then again, CGEL might consider only [ X and Y ] an NP-coordination, but not [ X, Y, and Z ]. – Damkerng T. – 2015-09-17T20:40:02.140
@DamkerngT. "Coordination is a relation between two or more elements of syntactically equal status, the coordinates; they are usually linked by means of a coordinator, such as and or or..." (Ch15 1.1)
– John B – 2015-09-17T21:36:04.5802
Thank you. I think that confirms my thought that [ X, Y, and Z ] can be considered an NP-coordination, and thus both is and are are possible in the OP's sentence. By the way, curious readers may find this Language Log post interesting: Agreement with nearest always bad?, There [ were (8) / was (8') ] liquor, music, and a strong desire for my body. -- Arnold Zwicky wrote "(8) has the "correct" (plural) agreement with expletive there, but it still sounds weird to me; I'd much prefer (8').
– Damkerng T. – 2015-09-17T22:00:23.370"There was Mary, Sue, Fred, Matt and Sam." is unbearably wrong to me. I can't accept it as correct, no matter what authoritative source says it's okay or right, and no matter what majority uses that construction. – ErikE – 2015-10-19T22:00:24.927
@DamkerngT. Thank you for pointing out footnote 21. You were right all along – John B – 2015-10-20T03:20:50.880