browser: `@vite/client` cannot load when a custom `base` is set

[nstepien]

Describe the bug

When a custom base is set in the vite config, browser tests fail to load /custom-base/@vite/client.
This error doesn't show up in the cli, but it does show up in the browser's console.
It doesn't seem to affect tests though, afaict.

Setting '/' for tests only fixes the issue.

const isTest = process.env.NODE_ENV === 'test';

export default defineConfig({
  base: isTest ? '/' : '/custom-base/',

Reproduction

export default defineConfig({
  base: '/custom-base/',
  // ...
  test: {
    browser: {
      enabled: true,
      name: 'chromium',
      provider: 'playwright',
      headless: false
    }
  }
});

Not minimal, but you can also reproduce with this repo:
https://github.com/adazzle/react-data-grid/blob/main/vite.config.ts#L11

System Info

System:
    OS: Windows 11 10.0.22631
    CPU: (64) x64 AMD Ryzen Threadripper 3970X 32-Core Processor
    Memory: 16.97 GB / 31.88 GB
  Binaries:
    Node: 22.8.0 - C:\Program Files\nodejs\node.EXE
    npm: 10.8.3 - C:\Program Files\nodejs\npm.CMD
  Browsers:
    Edge: Chromium (127.0.2651.74)
    Internet Explorer: 11.0.22621.3527
  npmPackages:
    @vitejs/plugin-react: ^4.3.1 => 4.3.1
    @vitest/browser: ^2.1.1 => 2.1.1
    @vitest/coverage-v8: ^2.1.1 => 2.1.1
    @vitest/eslint-plugin: ^1.1.4 => 1.1.4
    vite: ^5.4.5 => 5.4.5
    vitest: ^2.1.1 => 2.1.1

Used Package Manager

npm

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