BUG: inner join with StringDtype multi-indexes gives unexpected result in 1.5.x

[JeremyVriens]

Pandas version checks

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• 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

import pandas as pd

x = pd.DataFrame({"a": ["a1", "a1", "a2"], "b": ["b1", "b2", "b3"]}, dtype=pd.StringDtype())
x = x.set_index(["a", "b"])
y = pd.DataFrame({"a": ["a1"], "c": ["c1"]}, dtype=pd.StringDtype())
y = y.set_index(["a", "c"])
z = x.join(y, how="inner")

Issue Description

When joining 2 data frames with the StringDtype dtype index, the inner join gives incorrect results. The outcome of z returns a multi-index with only 1 row:
"a1", "b1", "c1"
It does work when I don't explicitly set the dtype. In that case the index dtypes are object.

Expected Behavior

The expected result for data frame z is 2 rows (just like in Pandas 1.4.x), namely:

"a1", "b1", "c1"
"a1", "b2", "c1"

Installed Versions

INSTALLED VERSIONS

commit : 91111fd
python : 3.10.8.final.0
python-bits : 64
OS : Windows
OS-release : 10
Version : 10.0.19044
machine : AMD64
processor : Intel64 Family 6 Model 85 Stepping 7, GenuineIntel
byteorder : little
LC_ALL : None
LANG : None
LOCALE : English_United States.1252
pandas : 1.5.1
numpy : 1.23.4
pytz : 2022.5
dateutil : 2.8.2
setuptools : 65.3.0
pip : 22.3
Cython : None
pytest : 7.1.3
hypothesis : None
sphinx : 5.3.0
blosc : None
feather : None
xlsxwriter : None
lxml.etree : None
html5lib : 1.1
pymysql : None
psycopg2 : None
jinja2 : 3.1.2
IPython : None
pandas_datareader: None
bs4 : None
bottleneck : None
brotli : 1.0.9
fastparquet : None
fsspec : None
gcsfs : None
matplotlib : None
numba : None
numexpr : None
odfpy : None
openpyxl : 3.0.10
pandas_gbq : None
pyarrow : None
pyreadstat : None
pyxlsb : None
s3fs : None
scipy : None
snappy : None
sqlalchemy : 1.4.42
tables : None
tabulate : 0.8.10
xarray : None
xlrd : None
xlwt : None
zstandard : None
tzdata : None

[JeremyVriens]

I couldn't really figure out which PR broke this, but maybe @jbrockmendel you know this? I've seen some merged PRs from you in 1.5.x that might be related to this.

[phofl]

commit 4248b23371a70b339a2c16b8e5caca9c2e5897f8
Author: jbrockmendel <jbrockmendel@gmail.com>
Date:   Tue Jan 25 09:31:24 2022 -0800

    ENH: ExtensionEngine (#45514)

cc @jbrockmendel #45514

[phofl]

This hits every ea dtype, not only string

[jbrockmendel]

@phofl im just catching up on an email backlog. does this still need my attention or do you have it covered in #49284?

[phofl]

This is covered, thangs for getting back to me