Cannot find WASI libc++ headers without normal libc++ headers installed

This bug concerns libc++-16-dev-wasm32 in Ubuntu mantic, but also applies to libc++-15-dev-wasm32 in Ubuntu lunar.

The problem can be seen in the following series of steps, starting with a minimal system:

  # apt-get install clang libc++-dev-wasm32

  $ echo '#include <cstring>' >test.cpp

  $ /usr/bin/clang++ --target=wasm32-wasi test.cpp -c
  tst.cpp:1:10: fatal error: 'cstring' file not found
  #include <cstring>
           ^~~~~~~~~
  1 error generated.

  # apt-get install libc++-dev

  $ /usr/bin/clang++ --target=wasm32-wasi test.cpp -c
  (success)

Note that Clang, with --target=wasm32-wasi, does not use any headers provided by libc++-16-dev; it uses the ones in libc++-16-dev-wasm32:

  $ /usr/bin/clang++ --target=wasm32-wasi test.cpp -c -H
  . /usr/include/wasm32-wasi/c++/v1/cstring
  .. /usr/include/wasm32-wasi/c++/v1/__assert
  ... /usr/include/wasm32-wasi/c++/v1/__config
  [...]

The headers provided by libc++-16-dev are extraneous to the requested compilation, so why is the compile affected by their absence/presence?

I encountered this issue in the course of attempting to build Firefox on the system. The following build log excerpt tells how this issue first arose:

  [...]
  checking the wasm C++ compiler works... yes
  checking the wasm C++ compiler can find wasi headers...
  DEBUG: Creating `/tmp/conftest.7wwkmmac.cpp` with content:
  DEBUG: | #include <cstring>
  DEBUG: | int
  DEBUG: | main(void)
  DEBUG: | {
  DEBUG: |
  DEBUG: | ;
  DEBUG: | return 0;
  DEBUG: | }
  DEBUG: Executing: `/usr/bin/clang++ --target=wasm32-wasi /tmp/conftest.7wwkmmac.cpp -c`
  DEBUG: The command returned non-zero exit status 1.
  DEBUG: Its error output was:
  DEBUG: | /tmp/conftest.7wwkmmac.cpp:1:10: fatal error: 'cstring' file not found
  DEBUG: | #include <cstring>
  DEBUG: | ^~~~~~~~~
  DEBUG: | 1 error generated.
  ERROR: Cannot find wasi headers or problem with the wasm compiler. Please fix the problem. Or build with --without-wasm-sandboxed-libraries.
  make[1]: *** [debian/rules:235: stamps/configure-browser] Error 1
  make[1]: Leaving directory '/home/build/firefox/firefox-116.0'
  make: *** [debian/rules:341: build] Error 2
  dpkg-buildpackage: error: debian/rules build subprocess returned exit status 2