diff --git a/tests/pip_install_upgrade/runtest.sh b/tests/pip_install_upgrade/runtest.sh
new file mode 100755
index 0000000..40e14a3
--- /dev/null
+++ b/tests/pip_install_upgrade/runtest.sh
@@ -0,0 +1,40 @@
+#!/bin/sh -eux
+# This script requires root privileges and you should never run it on your own machine
+test $EUID -eq 0
+
+PYTHON_VERSION=$(/usr/bin/python3 -c 'import sys; print("{}.{}".format(*sys.version_info))')
+RPM_SITELIB="/usr/lib/python${PYTHON_VERSION}/site-packages"
+LOCAL_SITELIB="/usr/local/lib/python${PYTHON_VERSION}/site-packages"
+USER_SITELIB="/home/fedora-test-user/.local/lib/python${PYTHON_VERSION}/site-packages"
+
+# First, let's install older Pello with pip as if it was installed by RPM
+# This is an approximation, but it usually works
+RPM_BUILD_ROOT=/ /usr/bin/pip install 'Pello==1.0.1'
+
+# Now, we'll upgrade it with regular pip
+/usr/bin/pip install --upgrade 'Pello==1.0.2'
+
+# pip should see it
+/usr/bin/pip freeze | grep '^Pello==1\.0\.2$'
+
+# Both installations should still exist
+test -d "${RPM_SITELIB}/Pello-1.0.1-py${PYTHON_VERSION}.egg-info" || test -d "${RPM_SITELIB}/Pello-1.0.1.dist-info"
+test -d "${LOCAL_SITELIB}/Pello-1.0.2.dist-info"
+
+# Let's ditch the local one
+/usr/bin/pip uninstall --yes Pello
+
+# It should only remove one of them
+test -d "${RPM_SITELIB}/Pello-1.0.1-py${PYTHON_VERSION}.egg-info" || test -d "${RPM_SITELIB}/Pello-1.0.1.dist-info"
+! test -d "${LOCAL_SITELIB}/Pello-1.0.2.dist-info"
+
+# And pip should still see the RPM-installed one
+/usr/bin/pip freeze | grep '^Pello==1\.0\.1$'
+
+# Again, but as regular user
+useradd fedora-test-user
+su fedora-test-user -c '/usr/bin/pip install "Pello==1.0.2"'
+test -d "${USER_SITELIB}/Pello-1.0.2.dist-info"
+su fedora-test-user -c '/usr/bin/pip freeze' | grep '^Pello==1\.0\.2$'
+su fedora-test-user -c '/usr/bin/pip uninstall --yes Pello'
+su fedora-test-user -c '/usr/bin/pip freeze' | grep '^Pello==1\.0\.1$'
diff --git a/tests/tests.yml b/tests/tests.yml
index b29f39e..fcbc430 100644
--- a/tests/tests.yml
+++ b/tests/tests.yml
@@ -63,6 +63,7 @@
     - pyproject_clikit:
         dir: pyproject-rpm-macros/tests
         run: ./mocktest.sh python-clikit
+    - pip_install_upgrade
     required_packages:
     - gcc
     - virtualenv
@@ -79,3 +80,6 @@
     - mock
     - rpmdevtools
     - rpm-build
+    - grep
+    - util-linux
+    - shadow-utils