BUG: `sort_values` is sorting the index too when `ignore_index=False`

[galipremsagar]

Pandas version checks

• I have checked that this issue has not already been reported.

• I have confirmed this bug exists on the latest version of pandas.

• I have confirmed this bug exists on the main branch of pandas.

Reproducible Example

In [1]: import pandas as pd

In [2]: pdf = pd.DataFrame({"a": [1, 3, 5, 2, 4], "b": [1, 1, 2, 2, 3], "c": [9, 7, 7, 7, 1]})

In [3]: pdf = pdf.set_index(['a', 'b'])

In [4]: pdf
Out[4]: 
     c
a b   
1 1  9
3 1  7
5 2  7
2 2  7
4 3  1

In [5]: pdf.sort_values(['c'], ignore_index=False)
Out[5]: 
     c
a b   
4 3  1
3 1  7
2 2  7
5 2  7
1 1  9

Issue Description

When there seems to be a tie in column c, the MultiIndex levels seem to be sorted - which is kind of unexpected. The expected behavior would be to preserve the order of occurrence right?

Expected Behavior

In [5]: pdf.sort_values(['c'], ignore_index=False)
Out[5]: 
     c
a b   
4 3  1
3 1  7
5 2  7
2 2  7
1 1  9

Installed Versions

INSTALLED VERSIONS

commit : fd3f571
python : 3.10.13.final.0
python-bits : 64
OS : Linux
OS-release : 5.15.0-94-generic
Version : #104-Ubuntu SMP Tue Jan 9 15:25:40 UTC 2024
machine : x86_64
processor : x86_64
byteorder : little
LC_ALL : None
LANG : en_US.UTF-8
LOCALE : en_US.UTF-8

pandas : 2.2.0
numpy : 1.26.4
pytz : 2024.1
dateutil : 2.8.2
setuptools : 69.1.0
pip : 24.0
Cython : 3.0.8
pytest : 7.4.4
hypothesis : 6.98.9
sphinx : 6.2.1
blosc : None
feather : None
xlsxwriter : None
lxml.etree : None
html5lib : None
pymysql : None
psycopg2 : None
jinja2 : 3.1.3
IPython : 8.21.0
pandas_datareader : None
adbc-driver-postgresql: None
adbc-driver-sqlite : None
bs4 : 4.12.3
bottleneck : None
dataframe-api-compat : None
fastparquet : None
fsspec : 2024.2.0
gcsfs : None
matplotlib : None
numba : 0.59.0
numexpr : None
odfpy : None
openpyxl : None
pandas_gbq : None
pyarrow : 14.0.1
pyreadstat : None
python-calamine : None
pyxlsb : None
s3fs : 2024.2.0
scipy : 1.12.0
sqlalchemy : 2.0.27
tables : None
tabulate : 0.9.0
xarray : None
xlrd : None
zstandard : None
tzdata : 2024.1
qtpy : None
pyqt5 : None

[mroeschke]

I cannot seemingly get the sorted result on main

In [1]: In [1]: import pandas as pd
   ...: 
   ...: In [2]: pdf = pd.DataFrame({"a": [1, 3, 5, 2, 4], "b": [1, 1, 2, 2, 3], "c": [9, 7, 7, 7, 1]})
   ...: 
   ...: In [3]: pdf = pdf.set_index(['a', 'b'])

In [2]: pdf.sort_values(['c'], ignore_index=False)
Out[2]: 
     c
a b   
4 3  1
3 1  7
5 2  7
2 2  7
1 1  9

But since the numpy sorts internally are unstable (xref #53558) maybe you're hitting a case where the result gets reordered?