Forgive me if this turns out to be a naive question. I'm quite convinced that not all abelian categories are (symmetric?) monoidal (of course, with the tensor product being additive), but I can't think of an abelian category that is not monoidal! Maybe I just know of too few abelian categories. Can someone give me an example?

Forgive me if this turns out to be a naive question. I'm quite convinced that not all abelian categories admit (symmetric?) monoidal structure (of course, with the tensor product being additive, meaning that it is an additive bifunctor), but I can't think of an abelian category that doesn't! Maybe I just know of too few abelian categories. Can someone give me an example?