diff --git a/news/7487.removal b/news/7487.removal
new file mode 100644
index 000000000..aa89bf42a
--- /dev/null
+++ b/news/7487.removal
@@ -0,0 +1,2 @@
+Wheel processing no longer permits wheels containing more than one top-level
+.dist-info directory.
diff --git a/src/pip/_internal/operations/install/wheel.py b/src/pip/_internal/operations/install/wheel.py
index 627cc4a32..4b10a9c00 100644
--- a/src/pip/_internal/operations/install/wheel.py
+++ b/src/pip/_internal/operations/install/wheel.py
@@ -315,8 +315,10 @@ def install_unpacked_wheel(
     :param pycompile: Whether to byte-compile installed Python files
     :param warn_script_location: Whether to check that scripts are installed
         into a directory on PATH
-    :raises UnsupportedWheel: when the directory holds an unpacked wheel with
-        incompatible Wheel-Version
+    :raises UnsupportedWheel:
+        * when the directory holds an unpacked wheel with incompatible
+          Wheel-Version
+        * when the .dist-info dir does not match the wheel
     """
     # TODO: Investigate and break this up.
     # TODO: Look into moving this into a dedicated class for representing an
@@ -381,8 +383,7 @@ def install_unpacked_wheel(
                     continue
                 elif (
                     is_base and
-                    s.endswith('.dist-info') and
-                    canonicalize_name(s).startswith(canonicalize_name(name))
+                    s.endswith('.dist-info')
                 ):
                     assert not info_dir, (
                         'Multiple .dist-info directories: {}, '.format(
@@ -445,6 +446,15 @@ def install_unpacked_wheel(
         req_description
     )
 
+    info_dir_name = canonicalize_name(os.path.basename(info_dir[0]))
+    canonical_name = canonicalize_name(name)
+    if not info_dir_name.startswith(canonical_name):
+        raise UnsupportedWheel(
+            "{} .dist-info directory {!r} does not start with {!r}".format(
+                req_description, os.path.basename(info_dir[0]), canonical_name
+            )
+        )
+
     # Get the defined entry points
     ep_file = os.path.join(info_dir[0], 'entry_points.txt')
     console, gui = get_entrypoints(ep_file)
diff --git a/tests/functional/test_install_wheel.py b/tests/functional/test_install_wheel.py
index e218d6e7f..4c1a92854 100644
--- a/tests/functional/test_install_wheel.py
+++ b/tests/functional/test_install_wheel.py
@@ -1,9 +1,11 @@
 import distutils
 import glob
 import os
+import shutil
 
 import pytest
 
+from tests.lib import create_basic_wheel_for_package
 from tests.lib.path import Path
 
 
@@ -432,3 +434,41 @@ def test_wheel_install_with_no_cache_dir(script, tmpdir, data):
     package = data.packages.joinpath("simple.dist-0.1-py2.py3-none-any.whl")
     result = script.pip('install', '--no-cache-dir', '--no-index', package)
     result.assert_installed('simpledist', editable=False)
+
+
+def test_wheel_install_fails_with_extra_dist_info(script):
+    package = create_basic_wheel_for_package(
+        script,
+        "simple",
+        "0.1.0",
+        extra_files={
+            "unrelated-2.0.0.dist-info/WHEEL": "Wheel-Version: 1.0",
+            "unrelated-2.0.0.dist-info/METADATA": (
+                "Name: unrelated\nVersion: 2.0.0\n"
+            ),
+        },
+    )
+    result = script.pip(
+        "install", "--no-cache-dir", "--no-index", package, expect_error=True
+    )
+    assert "Multiple .dist-info directories" in result.stderr
+
+
+def test_wheel_install_fails_with_unrelated_dist_info(script):
+    package = create_basic_wheel_for_package(script, "simple", "0.1.0")
+    new_name = "unrelated-2.0.0-py2.py3-none-any.whl"
+    new_package = os.path.join(os.path.dirname(package), new_name)
+    shutil.move(package, new_package)
+
+    result = script.pip(
+        "install",
+        "--no-cache-dir",
+        "--no-index",
+        new_package,
+        expect_error=True,
+    )
+
+    assert (
+        "'simple-0.1.0.dist-info' does not start with 'unrelated'"
+        in result.stderr
+    )
diff --git a/tests/lib/__init__.py b/tests/lib/__init__.py
index 07cc0ed5f..af80078df 100644
--- a/tests/lib/__init__.py
+++ b/tests/lib/__init__.py
@@ -921,8 +921,9 @@ def create_test_package_with_setup(script, **setup_kwargs):
     return pkg_path
 
 
-def create_basic_wheel_for_package(script, name, version,
-                                   depends=None, extras=None):
+def create_basic_wheel_for_package(
+    script, name, version, depends=None, extras=None, extra_files=None
+):
     if depends is None:
         depends = []
     if extras is None:
@@ -966,6 +967,9 @@ def create_basic_wheel_for_package(script, name, version,
         "{dist_info}/RECORD": ""
     }
 
+    if extra_files:
+        files.update(extra_files)
+
     # Some useful shorthands
     archive_name = "{name}-{version}-py2.py3-none-any.whl".format(
         name=name, version=version