Trainer initialization freezes when mpi4py is installed

Bug description

Trainer freezes on initialization when mpi4py is installed.

I suspect the following issues are encountering the same problem: #18836 #19768

What version are you seeing the problem on?

master

How to reproduce the bug

This freezes

pip install lightning mpi4py
python -c "import lightning; lightning.Trainer(accelerator='cpu')"

This does not freeze

pip install lightning
python -c "import lightning; lightning.Trainer(accelerator='cpu')"

Error messages and logs

No response

Environment

Conda environment that freezes:

name: debuglightning
channels:
  - conda-forge
  - defaults
dependencies:
  - _libgcc_mutex=0.1=conda_forge
  - _openmp_mutex=4.5=2_gnu
  - bzip2=1.0.8=hd590300_5
  - ca-certificates=2024.7.4=hbcca054_0
  - ld_impl_linux-64=2.40=hf3520f5_7
  - libexpat=2.6.2=h59595ed_0
  - libffi=3.4.2=h7f98852_5
  - libgcc-ng=14.1.0=h77fa898_0
  - libgomp=14.1.0=h77fa898_0
  - libnsl=2.0.1=hd590300_0
  - libsqlite=3.46.0=hde9e2c9_0
  - libuuid=2.38.1=h0b41bf4_0
  - libxcrypt=4.4.36=hd590300_1
  - libzlib=1.3.1=h4ab18f5_1
  - ncurses=6.5=h59595ed_0
  - openssl=3.3.1=h4ab18f5_1
  - pip=24.0=pyhd8ed1ab_0
  - python=3.11.9=hb806964_0_cpython
  - readline=8.2=h8228510_1
  - setuptools=70.1.1=pyhd8ed1ab_0
  - tk=8.6.13=noxft_h4845f30_101
  - tzdata=2024a=h0c530f3_0
  - wheel=0.43.0=pyhd8ed1ab_1
  - xz=5.2.6=h166bdaf_0
  - pip:
      - aiohttp==3.9.5
      - aiosignal==1.3.1
      - attrs==23.2.0
      - filelock==3.15.4
      - frozenlist==1.4.1
      - fsspec==2024.6.1
      - idna==3.7
      - jinja2==3.1.4
      - lightning==2.3.2
      - lightning-utilities==0.11.3.post0
      - markupsafe==2.1.5
      - mpi4py==3.1.6
      - mpmath==1.3.0
      - multidict==6.0.5
      - networkx==3.3
      - numpy==2.0.0
      - nvidia-cublas-cu12==12.1.3.1
      - nvidia-cuda-cupti-cu12==12.1.105
      - nvidia-cuda-nvrtc-cu12==12.1.105
      - nvidia-cuda-runtime-cu12==12.1.105
      - nvidia-cudnn-cu12==8.9.2.26
      - nvidia-cufft-cu12==11.0.2.54
      - nvidia-curand-cu12==10.3.2.106
      - nvidia-cusolver-cu12==11.4.5.107
      - nvidia-cusparse-cu12==12.1.0.106
      - nvidia-nccl-cu12==2.20.5
      - nvidia-nvjitlink-cu12==12.5.82
      - nvidia-nvtx-cu12==12.1.105
      - packaging==24.1
      - pytorch-lightning==2.3.2
      - pyyaml==6.0.1
      - sympy==1.12.1
      - torch==2.3.1
      - torchmetrics==1.4.0.post0
      - tqdm==4.66.4
      - triton==2.3.1
      - typing-extensions==4.12.2
      - yarl==1.9.4
prefix: /home/robinysh/.conda/envs/debuglightning

cc @awaelchli

Start with the lightning.Trainer entry point and reproduce the freeze using the provided Python command after installing mpi4py, then compare it with the environment without mpi4py. Trace initialization until the hang is identified; done means Trainer initialization completes on CPU with mpi4py installed and the existing no-mpi4py case still works.