diff --git a/.github/workflows/twisted_trunk.yml b/.github/workflows/twisted_trunk.yml
index 5f0671f350..12267405be 100644
--- a/.github/workflows/twisted_trunk.yml
+++ b/.github/workflows/twisted_trunk.yml
@@ -96,6 +96,72 @@ jobs:
             /logs/results.tap
             /logs/**/*.log*
 
+  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:
+      # The path is set via a file given by $GITHUB_PATH. We need both Go 1.17 and GOPATH on the path to run Complement.
+      # See https://docs.github.com/en/actions/using-workflows/workflow-commands-for-github-actions#adding-a-system-path
+      - name: "Set Go Version"
+        run: |
+          # Add Go 1.17 to the PATH: see https://github.com/actions/virtual-environments/blob/main/images/linux/Ubuntu2004-Readme.md#environment-variables-2
+          echo "$GOROOT_1_17_X64/bin" >> $GITHUB_PATH
+          # Add the Go path to the PATH: We need this so we can call gotestfmt
+          echo "~/go/bin" >> $GITHUB_PATH
+
+      - name: "Install Complement Dependencies"
+        run: |
+          sudo apt-get update && sudo apt-get install -y libolm3 libolm-dev
+          go get -v github.com/haveyoudebuggedit/gotestfmt/v2/cmd/gotestfmt@latest
+
+      - name: Run actions/checkout@v2 for synapse
+        uses: actions/checkout@v2
+        with:
+          path: synapse
+
+      # This step is specific to the 'Twisted trunk' test run:
+      - name: Patch dependencies
+        run: |
+          set -x
+          DEBIAN_FRONTEND=noninteractive sudo apt-get install -yqq python3 pipx
+          pipx install poetry==1.1.12
+
+          poetry remove -n twisted
+          poetry add -n --extras tls git+https://github.com/twisted/twisted.git#trunk
+          poetry lock --no-update
+          # NOT IN 1.1.12 poetry lock --check
+        working-directory: synapse
+
+      - name: "Install custom gotestfmt template"
+        run: |
+          mkdir .gotestfmt/github -p
+          cp synapse/.ci/complement_package.gotpl .gotestfmt/github/package.gotpl
+
+      # Attempt to check out the same branch of Complement as the PR. If it
+      # doesn't exist, fallback to HEAD.
+      - name: Checkout complement
+        run: synapse/.ci/scripts/checkout_complement.sh
+
+      - run: |
+          set -o pipefail
+          TEST_ONLY_SKIP_DEP_HASH_VERIFICATION=1 POSTGRES=${{ (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:
     if: failure()
@@ -103,6 +169,7 @@ jobs:
       - mypy
       - trial
       - sytest
+      - complement
 
     runs-on: ubuntu-latest
 
diff --git a/changelog.d/13079.misc b/changelog.d/13079.misc
new file mode 100644
index 0000000000..0133097c83
--- /dev/null
+++ b/changelog.d/13079.misc
@@ -0,0 +1 @@
+Enable Complement testing in the 'Twisted Trunk' CI runs.
\ No newline at end of file
diff --git a/docker/Dockerfile b/docker/Dockerfile
index c676f83775..22707ed142 100644
--- a/docker/Dockerfile
+++ b/docker/Dockerfile
@@ -62,7 +62,13 @@ WORKDIR /synapse
 # Copy just what we need to run `poetry export`...
 COPY pyproject.toml poetry.lock /synapse/
 
-RUN /root/.local/bin/poetry export --extras all -o /synapse/requirements.txt
+
+# If specified, we won't verify the hashes of dependencies.
+# This is only needed if the hashes of dependencies cannot be checked for some
+# 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}
 
 ###
 ### Stage 1: builder
@@ -85,6 +91,7 @@ RUN \
     openssl \
     rustc \
     zlib1g-dev \
+    git \
     && rm -rf /var/lib/apt/lists/*
 
 # To speed up rebuilds, install all of the dependencies before we copy over
diff --git a/scripts-dev/complement.sh b/scripts-dev/complement.sh
index 20df5fbc24..8448d49e26 100755
--- a/scripts-dev/complement.sh
+++ b/scripts-dev/complement.sh
@@ -23,6 +23,9 @@
 #
 # ./complement.sh -run "TestOutboundFederation(Profile|Send)"
 #
+# Specifying TEST_ONLY_SKIP_DEP_HASH_VERIFICATION=1 will cause `poetry export`
+# to not emit any hashes when building the Docker image. This then means that
+# you can use 'unverifiable' sources such as git repositories as dependencies.
 
 # Exit if a line returns a non-zero exit code
 set -e
@@ -86,7 +89,9 @@ fi
 if [ -z "$skip_docker_build" ]; then
     # Build the base Synapse image from the local checkout
     echo_if_github "::group::Build Docker image: matrixdotorg/synapse"
-    docker build -t matrixdotorg/synapse -f "docker/Dockerfile" .
+    docker build -t matrixdotorg/synapse \
+      --build-arg TEST_ONLY_SKIP_DEP_HASH_VERIFICATION \
+      -f "docker/Dockerfile" .
     echo_if_github "::endgroup::"
 
     # Build the workers docker image (from the base Synapse image we just built).