@markmccaughrean The latter bit about word order is very hard to explain and definitely something "you just know". English isn't unique that way. Lots of languages have things more felt than easy to explain.
I think with the present there - it puts a slightly different emphasis. And it works in this case because you can "have fish" as a simple thing...it doesn't suffer from lack of emphasis on it being a continuing present thing. You can also say "we have friends over" and match the two tenses