Rust does not have native C++ interop.
C ABI is different to what clasp has done (native C++ interop). All widely used Common Lisp implementations have seamless C ABI interop through their FFIs / CFFI [1].
Clasp goes above and beyond that and natively interfaces with C++. LTO works with mixed Common Lisp and C++ code. Exception handling works as expected. You can mix Common Lisp and C++ stack frames. Correct me if I'm wrong but Rust does not have anything equivalent. The language that comes closest is D I think [2].