`to_netcdf` fails to write to files previously opened with `open_mfdataset`

What happened?

I've been working in a system where I occasionally read a multifile dataset and, in effect within the same Python process, subsequently want to write to some of the files containing that dataset. Instead, I get permission errors when I try to write to the file.

What did you expect to happen?

I expected to not get a permission error.

Minimal Complete Verifiable Example

import xarray as xr

ds = xr.Dataset(
    attrs={'blah': -1},
    coords=({'x': range(10)}),
    data_vars={'y': ('x', [10*i for i in range(10)])},
)

ds.isel(x=slice(0,5)).to_netcdf('file1.nc')
ds.isel(x=slice(5,10)).to_netcdf('file2.nc')

a = xr.open_mfdataset(paths=['file1.nc', 'file2.nc'])

ds.to_netcdf('file1.nc')

MVCE confirmation

• Minimal example — the example is as focused as reasonably possible to demonstrate the underlying issue in xarray.
• Complete example — the example is self-contained, including all data and the text of any traceback.
• Verifiable example — the example copy & pastes into an IPython prompt or Binder notebook, returning the result.
• New issue — a search of GitHub Issues suggests this is not a duplicate.
• Recent environment — the issue occurs with the latest version of xarray and its dependencies.

Relevant log output

Traceback (most recent call last):
  File "/home/wball/.local/lib/python3.12/site-packages/xarray/backends/file_manager.py", line 211, in _acquire_with_cache_info
    file = self._cache[self._key]
           ~~~~~~~~~~~^^^^^^^^^^^
  File "/home/wball/.local/lib/python3.12/site-packages/xarray/backends/lru_cache.py", line 56, in __getitem__
    value = self._cache[key]
            ~~~~~~~~~~~^^^^^
KeyError: [<class 'netCDF4._netCDF4.Dataset'>, ('/home/wball/tmp/file1.nc',), 'a', (('clobber', True), ('diskless', False), ('format', 'NETCDF4'), ('persist', False)), '11d144ce-327c-44e3-82f5-fb97c2d66e65']

During handling of the above exception, another exception occurred:

Traceback (most recent call last):
  File "/home/wball/tmp/xarray-mwe.py", line 14, in <module>
    ds.to_netcdf('file1.nc')
  File "/home/wball/.local/lib/python3.12/site-packages/xarray/core/dataset.py", line 2329, in to_netcdf
    return to_netcdf(  # type: ignore  # mypy cannot resolve the overloads:(
           ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
  File "/home/wball/.local/lib/python3.12/site-packages/xarray/backends/api.py", line 1343, in to_netcdf
    store = store_open(target, mode, format, group, **kwargs)
            ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
  File "/home/wball/.local/lib/python3.12/site-packages/xarray/backends/netCDF4_.py", line 408, in open
    return cls(manager, group=group, mode=mode, lock=lock, autoclose=autoclose)
           ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
  File "/home/wball/.local/lib/python3.12/site-packages/xarray/backends/netCDF4_.py", line 355, in __init__
    self.format = self.ds.data_model
                  ^^^^^^^
  File "/home/wball/.local/lib/python3.12/site-packages/xarray/backends/netCDF4_.py", line 417, in ds
    return self._acquire()
           ^^^^^^^^^^^^^^^
  File "/home/wball/.local/lib/python3.12/site-packages/xarray/backends/netCDF4_.py", line 411, in _acquire
    with self._manager.acquire_context(needs_lock) as root:
  File "/usr/lib64/python3.12/contextlib.py", line 137, in __enter__
    return next(self.gen)
           ^^^^^^^^^^^^^^
  File "/home/wball/.local/lib/python3.12/site-packages/xarray/backends/file_manager.py", line 199, in acquire_context
    file, cached = self._acquire_with_cache_info(needs_lock)
                   ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
  File "/home/wball/.local/lib/python3.12/site-packages/xarray/backends/file_manager.py", line 217, in _acquire_with_cache_info
    file = self._opener(*self._args, **kwargs)
           ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
  File "src/netCDF4/_netCDF4.pyx", line 2469, in netCDF4._netCDF4.Dataset.__init__
  File "src/netCDF4/_netCDF4.pyx", line 2028, in netCDF4._netCDF4._ensure_nc_success
PermissionError: [Errno 13] Permission denied: '/home/wball/tmp/file1.nc'

Anything else we need to know?

No response

Environment

Start by reproducing the example and reading xarray/backends/file_manager.py and xarray/backends/netCDF4_.py, the paths shown in the traceback. Trace how open_mfdataset retains file handles before the later to_netcdf call. Done means the example can reopen file1.nc for writing without a permission error, with regression coverage for this sequence.