Which of the following sentences is correct:
- There is a lot of food and fruit in the supermarket.
- There are a lot of food and fruit in the supermarket.
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I wouldn't use either. Would you say 'I met a lot of sportsmen and baseball players yesterday'?
And with a better example, I'd be careful:
With mass nouns, where the coordinated nouns are unitary or notionally closely linked,
'There is a lot of trouble and strife in the world'
'There is a lot of rice and barley on the shelves'
(compound modifier, compare 'There is plenty of rice and barley on the shelves').
but with count nouns
'There are lots / a lot of men and women in the street'.
The first thing to consider is that a lot of and lots of are what CGEL calls 'number-transparent quantifi[ers]': essentially, forget that 'lot/s' was once a noun and treat as a MWQ meaning much/many. It is the quantified noun group (rice, rice and barley, trouble and strife) that determines verb agreement, not whether lot has the s.
Secondly, however, it is not always obvious what is the preferred agreement for a coordinated noun group:
bacon and eggs is my favourite food / there is a lot of bacon and eggs on my plate / lots of rice and barley was found in the pantry
bacon and eggs have both gone up enormously over the last few years / a lot of bacon and a lot of eggs were stored in the fridge.
The first one is correct: There is a lot of food and fruit in the supermarket.
Since you are using the adjectival phrase "a lot of...", this makes it singular no matter the number of items in the set. Food and fruit are a pluralized noun set without an initial adjectival phrase. Oh look, I just used it in an example.
Another way to see it is to change the sentence: There are lots of food and fruit in the supermarket.