Is there a generally accepted alternative to GMP for arbitrary precision?

The implementation of the Tcl language uses libtommath for its bigint library, and it does indeed appear to be suitable for production use. (It's a C library, not C++, but it's certainly suitable.)

Well, as a large project like GCC uses this for its printf and compile-time calculations (correct me if I'm wrong here), I think it's stable enough. I would think there are ways to limit the out-of-memory errors the commenter worries about, but I haven't used it personally, so I can't help you there. All I'm saying is that it is a proven stable library. It even has a C++ interface library.