The sentence
Here is your umbrella and your coat.
is from New Concept English. So I presume it is grammatically correct.
I also saw sentences like
There is an apple and two oranges.
So why is is used instead of are?
The sentence
Here is your umbrella and your coat.
is from New Concept English. So I presume it is grammatically correct.
I also saw sentences like
There is an apple and two oranges.
So why is is used instead of are?
The rule I learnt is that the verb / auxiliary agrees with the first member of the list. E.g.:
There is a black cat, a white cat, and two ginger cats.
There are four cats. One is black, one is white, and two are ginger.
There is is coming to be used as an invariable term to introduce items whether they are singular or plural. The same may also be happening with here is, but in your example 'your umbrella and your coat' can be regarded as a single item, in so far as the two are being offered together.
In spite of what grammar rules may say, as other answers have noted, "there is" is a popular way to introduce items singular or plural. For your particular examples, my explanation for the extent of their acceptance is that they are understood as elliptic forms:
• Here is your umbrella and your coat
← Here is your umbrella, and here is your coat
• Here is an apple and two oranges
← Here is an apple, and here are two oranges
I agree with jwpat7 about the understanding with what's left off at the end and implied. The "and" truncates what could be a longer sentence as in that answer above (an elliptic form). You could stop the sentence, in other words, at "There is an apple." Or have another sentence "There are two oranges." But combined together, "There is an apple and two oranges" is correct because it could continue as "There is an apple and two oranges on the table" and it sounds correct. We know it is right since we can remove the phrase "and two oranges" and the sentence still works. Two commas could be placed around the phrase even since it's almost a side note now in my longer sentence: "There is an apple, and two oranges, on the table."