diff --git a/.github/workflows/latest_deps.yml b/.github/workflows/latest_deps.yml
index c9ec70abe..cb801afcb 100644
--- a/.github/workflows/latest_deps.yml
+++ b/.github/workflows/latest_deps.yml
@@ -197,11 +197,14 @@ jobs:
         with:
           path: synapse
 
-      - uses: actions/setup-go@v4
-
       - name: Prepare Complement's Prerequisites
         run: synapse/.ci/scripts/setup_complement_prerequisites.sh
 
+      - uses: actions/setup-go@v4
+        with:
+          cache-dependency-path: complement/go.sum
+          go-version-file: complement/go.mod
+
       - 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 | synapse/.ci/scripts/gotestfmt
diff --git a/.github/workflows/tests.yml b/.github/workflows/tests.yml
index 12420911b..a1f714da2 100644
--- a/.github/workflows/tests.yml
+++ b/.github/workflows/tests.yml
@@ -633,11 +633,14 @@ jobs:
         uses: dtolnay/rust-toolchain@1.61.0
       - uses: Swatinem/rust-cache@v2
 
-      - uses: actions/setup-go@v4
-
       - name: Prepare Complement's Prerequisites
         run: synapse/.ci/scripts/setup_complement_prerequisites.sh
 
+      - uses: actions/setup-go@v4
+        with:
+          cache-dependency-path: complement/go.sum
+          go-version-file: complement/go.mod
+
         # use p=1 concurrency as GHA boxes are underpowered and don't like running tons of synapses at once.
       - run: |
           set -o pipefail
diff --git a/.github/workflows/twisted_trunk.yml b/.github/workflows/twisted_trunk.yml
index 062f782e8..1011a1539 100644
--- a/.github/workflows/twisted_trunk.yml
+++ b/.github/workflows/twisted_trunk.yml
@@ -168,11 +168,14 @@ jobs:
         with:
           path: synapse
 
-      - uses: actions/setup-go@v4
-
       - name: Prepare Complement's Prerequisites
         run: synapse/.ci/scripts/setup_complement_prerequisites.sh
 
+      - uses: actions/setup-go@v4
+        with:
+          cache-dependency-path: complement/go.sum
+          go-version-file: complement/go.mod
+
       # This step is specific to the 'Twisted trunk' test run:
       - name: Patch dependencies
         run: |
diff --git a/changelog.d/16567.misc b/changelog.d/16567.misc
new file mode 100644
index 000000000..858fbac7f
--- /dev/null
+++ b/changelog.d/16567.misc
@@ -0,0 +1 @@
+Deal with warnings from running complement in CI.