diff --git a/.github/workflows/latest_deps.yml b/.github/workflows/latest_deps.yml
index c537a5a60..f263cf612 100644
--- a/.github/workflows/latest_deps.yml
+++ b/.github/workflows/latest_deps.yml
@@ -135,11 +135,42 @@ jobs:
             /logs/**/*.log*
 
 
-  # TODO: run complement (as with twisted trunk, see #12473).
+  complement:
+    if: "${{ !failure() && !cancelled() }}"
+    runs-on: ubuntu-latest
 
-  # open an issue if the build fails, so we know about it.
+    strategy:
+      fail-fast: false
+      matrix:
+        include:
+          - arrangement: monolith
+            database: SQLite
+
+          - arrangement: monolith
+            database: Postgres
+
+          - arrangement: workers
+            database: Postgres
+
+    steps:
+      - name: Run actions/checkout@v2 for synapse
+        uses: actions/checkout@v2
+        with:
+          path: synapse
+
+      - name: Prepare Complement's Prerequisites
+        run: synapse/.ci/scripts/setup_complement_prerequisites.sh
+
+      - run: |
+          set -o pipefail
+          TEST_ONLY_IGNORE_POETRY_LOCKFILE=1 POSTGRES=${{ (matrix.database == 'Postgres') && 1 || '' }} WORKERS=${{ (matrix.arrangement == 'workers') && 1 || '' }} COMPLEMENT_DIR=`pwd`/complement synapse/scripts-dev/complement.sh -json 2>&1 | gotestfmt
+        shell: bash
+        name: Run Complement Tests
+
+  # Open an issue if the build fails, so we know about it.
+  # Only do this if we're not experimenting with this action in a PR.
   open-issue:
-    if: failure()
+    if: "failure() && github.event_name != 'push' && github.event_name != 'pull_request'"
     needs:
       # TODO: should mypy be included here? It feels more brittle than the other two.
       - mypy
diff --git a/changelog.d/13213.misc b/changelog.d/13213.misc
new file mode 100644
index 000000000..b50d26ac0
--- /dev/null
+++ b/changelog.d/13213.misc
@@ -0,0 +1 @@
+Enable Complement CI tests in the 'latest deps' test run.
\ No newline at end of file
diff --git a/docker/Dockerfile b/docker/Dockerfile
index f4d8e6c92..97bb03b08 100644
--- a/docker/Dockerfile
+++ b/docker/Dockerfile
@@ -68,7 +68,18 @@ COPY pyproject.toml poetry.lock /synapse/
 # reason, such as when a git repository is used directly as a dependency.
 ARG TEST_ONLY_SKIP_DEP_HASH_VERIFICATION
 
-RUN /root/.local/bin/poetry export --extras all -o /synapse/requirements.txt ${TEST_ONLY_SKIP_DEP_HASH_VERIFICATION:+--without-hashes}
+# If specified, we won't use the Poetry lockfile.
+# Instead, we'll just install what a regular `pip install` would from PyPI.
+ARG TEST_ONLY_IGNORE_POETRY_LOCKFILE
+
+# Export the dependencies, but only if we're actually going to use the Poetry lockfile.
+# Otherwise, just create an empty requirements file so that the Dockerfile can
+# proceed.
+RUN if [ -z "$TEST_ONLY_IGNORE_POETRY_LOCKFILE" ]; then \
+    /root/.local/bin/poetry export --extras all -o /synapse/requirements.txt ${TEST_ONLY_SKIP_DEP_HASH_VERIFICATION:+--without-hashes}; \
+  else \
+    touch /synapse/requirements.txt; \
+  fi
 
 ###
 ### Stage 1: builder
@@ -108,8 +119,17 @@ COPY synapse /synapse/synapse/
 # ... and what we need to `pip install`.
 COPY pyproject.toml README.rst /synapse/
 
+# Repeat of earlier build argument declaration, as this is a new build stage.
+ARG TEST_ONLY_IGNORE_POETRY_LOCKFILE
+
 # Install the synapse package itself.
-RUN pip install --prefix="/install" --no-deps --no-warn-script-location /synapse
+# If we have populated requirements.txt, we don't install any dependencies
+# as we should already have those from the previous `pip install` step.
+RUN if [ -z "$TEST_ONLY_IGNORE_POETRY_LOCKFILE" ]; then \
+    pip install --prefix="/install" --no-deps --no-warn-script-location /synapse[all]; \
+  else \
+    pip install --prefix="/install" --no-warn-script-location /synapse[all]; \
+  fi
 
 ###
 ### Stage 2: runtime
diff --git a/scripts-dev/complement.sh b/scripts-dev/complement.sh
index 6381f7092..eab23f18f 100755
--- a/scripts-dev/complement.sh
+++ b/scripts-dev/complement.sh
@@ -101,6 +101,7 @@ if [ -z "$skip_docker_build" ]; then
     echo_if_github "::group::Build Docker image: matrixdotorg/synapse"
     docker build -t matrixdotorg/synapse \
       --build-arg TEST_ONLY_SKIP_DEP_HASH_VERIFICATION \
+      --build-arg TEST_ONLY_IGNORE_POETRY_LOCKFILE \
       -f "docker/Dockerfile" .
     echo_if_github "::endgroup::"