The combination might seem more than a bit awkward at first, but I think I've found a title that combines both series perfectly. And while, objectively, it's more of a "Matrix done right" than anything like Silent Hill, the series in question somehow felt much closer to Silent Hill 2 than anything I've previously seen.
I'm talking about Serial Experiments Lain (don't ask me what the title means, because I have no idea). It's a 13-episode anime series from 1998, which tells a story of a young girl called Lain. At first the story is very similar to the one of Matrix, minus kung-fu and gravity-defying stunts. Nothing surprising, considering Lain's constitution. However, the series quickly delves into the inner world of its lead character, whose journey into the Wired (a mix between Internet and Matrix) becomes quite similar to James' journey into Silent Hill. Somehow, Lain is also very similar to James, with both of them unwilling to accept something they know is true. In the later episodes, reality is twisted in a way that allows the use of some impressive and surrealistic psychological horror elements. And that sure is lovely, don't you think?
Overall, I think the series was definitely one of the best serious titles I've ever seen, anime or not. If you like sci-fi and psychological horror and don't mind the complete absence of light-hearted elements, you should definitely give it a try.