Route handler does not receive searchParams set by middleware without runtime edge

Link to the code that reproduces this issue

https://github.com/LuukSantegoeds/nextjs-searchparams-issue

To Reproduce

Within the linked repository:

1. Navigate to http://localhost:3000/route-default?foo=bar (default runtime) and validate that the searchParam is received by the route by checking the json response
2. Navigate to http://localhost:3000/route-edge?foo=bar (edge runtime) and validate that the searchParam is received by the route by checking the json response
3. Navigate to http://localhost:3000/middleware/route-edge?bar=foo (edge runtime) and validate that a searchParam is added by the middleware to the searchParam already provided
4. Navigate to http://localhost:3000/middleware/route-default?bar=foo where no will see that no additional searchParam is added by the middleware

Current vs. Expected behavior

I would expect that the searchParam added by the middleware (in case of the default runtime) would also be received by the route handler, just like with the edge runtime.

This only occurs in combination with a rewrite in the middleware. I you were to disable the middleware and do direct requests to the route handler, searchParams are received in both runtime situations as demonstrated within the first 2 steps.

Provide environment information

Operating System:
  Platform: win32
  Arch: x64
  Version: Windows 11 Enterprise
  Available memory (MB): 32488
  Available CPU cores: 16
Binaries:
  Node: 20.9.0
  npm: 10.1.0
  Yarn: N/A
  pnpm: 8.15.7
Relevant Packages:
  next: 14.2.2 // Latest available version is detected (14.2.2).
  eslint-config-next: 14.2.2
  react: 18.2.0
  react-dom: 18.2.0
  typescript: 5.4.5
Next.js Config:
  output: N/A

Which area(s) are affected? (Select all that apply)

Middleware & route handlers

Which stage(s) are affected? (Select all that apply)

next dev (local), next start (local)

Additional context

This was tested locally

Start with the linked reproduction repository and run the four listed URLs to compare default and edge runtimes. Inspect the middleware rewrite and the route handlers used by those paths. Done means the default-runtime handler receives the search parameter added by middleware, matching edge-runtime behavior, with a regression test covering the case.