Enable Complement tests against latest deps

.github/workflows/latest_deps.yml

@@ -135,7 +135,37 @@ jobs:
             /logs/**/*.log*
 
 
-  # TODO: run complement (as with twisted trunk, see #12473).
+  complement:
+    if: "${{ !failure() && !cancelled() }}"
+    runs-on: ubuntu-latest
+
+    strategy:
+      fail-fast: false
+      matrix:
+        include:
+          - arrangement: monolith
+            database: SQLite
+
+          - arrangement: monolith
+            database: Postgres
+
+          - arrangement: workers
+            database: Postgres
+
+    steps:
+      - name: Run actions/checkout@v2 for synapse
+        uses: actions/checkout@v2
+        with:
+          path: synapse
+
+      - name: Prepare Complement's Prerequisites
+        run: synapse/.ci/scripts/setup_complement_prerequisites.sh
+
+      - run: |
+          set -o pipefail
+          TEST_ONLY_IGNORE_POETRY_LOCKFILE=1 POSTGRES=${{ (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.
   open-issue: