diff --git a/README.md b/README.md
index ef7347b..9eebfa4 100644
--- a/README.md
+++ b/README.md
@@ -347,6 +347,12 @@ are outside `make check`. Neither opens the popup from the toolbar button: both
 load its page in an ordinary tab, so what the toolbar popup itself adds, its
 size and its closing when it loses focus, is covered by neither.
 
+Each run builds its image under a tag of its own, `autistmask-e2e-chrome:` or
+`autistmask-e2e-firefox:` followed by twelve random hex digits, and removes that
+image when it exits, an interrupt included. Two clones can therefore run the
+same suite at once without one build taking the other's image away, and no run
+leaves an image behind.
+
 ### Chrome (`make test-e2e`)
 
 `make test-e2e` builds `dist/chrome/` and drives the **real popup in a real
diff --git a/script/test-e2e b/script/test-e2e
index 2246905..43a3450 100755
--- a/script/test-e2e
+++ b/script/test-e2e
@@ -22,13 +22,18 @@ set -eu
 SCRIPT_DIR="$(cd "$(dirname "$0")" && pwd -P)"
 ROOT="$(cd "$SCRIPT_DIR/.." && pwd -P)"
 
-IMAGE="$("$SCRIPT_DIR/projectname")-e2e-chrome"
-
-IIDFILE=""
+# Each run builds, runs and then removes an image under a tag of its own.
+# Under one tag shared by every run, a second clone's build moves the tag
+# off this run's image, and the image store can then delete that untagged
+# image before or while this run uses it. The tag is random rather than
+# $$ because CI jobs share one docker daemon, but each job's container
+# numbers its processes from scratch.
+IMAGE="$("$SCRIPT_DIR/projectname")-e2e-chrome:$(od -An -N6 -tx1 /dev/urandom | tr -d ' \n')"
 
 cleanup() {
-    if [ -n "$IIDFILE" ]; then
-        rm -f "$IIDFILE"
+    # No image to remove when the build failed or was interrupted.
+    if docker image inspect "$IMAGE" >/dev/null 2>&1; then
+        docker image rm "$IMAGE" >/dev/null
     fi
 }
 
@@ -40,22 +45,15 @@ main() {
         exit 1
     fi
 
-    IIDFILE="$(mktemp)"
     trap cleanup EXIT
     trap 'cleanup; exit 130' INT TERM
 
     echo "Building the Chrome e2e image (extension included)..."
     # --no-cache: the image build runs make build and its checks, and a
     # cached layer is a check that did not run.
-    docker build --no-cache --iidfile "$IIDFILE" -t "$IMAGE" \
-        -f tests/e2e/Dockerfile .
+    docker build --no-cache -t "$IMAGE" -f tests/e2e/Dockerfile .
 
     echo "Running e2e suite in the pinned Playwright container..."
-    # The image is run by ID, not by tag: where two clones of this repo run
-    # the suite at once, the other build can move the tag between this
-    # build and this run, and the suite would then silently test the other
-    # checkout.
-    #
     # --ipc=host: Chromium's shared-memory needs more than the default
     #   64MB /dev/shm or renderers crash.
     # HOME=/tmp: the image's root home is not a reliable place for the
@@ -79,7 +77,7 @@ main() {
         -e HOME=/tmp \
         -e PW_EXPERIMENTAL_SERVICE_WORKER_NETWORK_EVENTS=1 \
         -e "E2E_TRACE_NETWORK=${E2E_TRACE_NETWORK:-0}" \
-        "$(cat "$IIDFILE")" \
+        "$IMAGE" \
         node tests/e2e/run.js
 
     # Where chrome.storage.local lives, and what moves it: two unpacked loads
@@ -95,7 +93,7 @@ main() {
     docker run --rm \
         --ipc=host \
         -e HOME=/tmp \
-        "$(cat "$IIDFILE")" \
+        "$IMAGE" \
         node tests/e2e/storagePartition.js
 }
 
diff --git a/script/test-e2e-firefox b/script/test-e2e-firefox
index 892e7e1..6058ba3 100755
--- a/script/test-e2e-firefox
+++ b/script/test-e2e-firefox
@@ -21,13 +21,13 @@ set -eu
 SCRIPT_DIR="$(cd "$(dirname "$0")" && pwd -P)"
 ROOT="$(cd "$SCRIPT_DIR/.." && pwd -P)"
 
-IMAGE="$("$SCRIPT_DIR/projectname")-e2e-firefox"
-
-IIDFILE=""
+# A tag of this run's own, as in script/test-e2e, which says why.
+IMAGE="$("$SCRIPT_DIR/projectname")-e2e-firefox:$(od -An -N6 -tx1 /dev/urandom | tr -d ' \n')"
 
 cleanup() {
-    if [ -n "$IIDFILE" ]; then
-        rm -f "$IIDFILE"
+    # No image to remove when the build failed or was interrupted.
+    if docker image inspect "$IMAGE" >/dev/null 2>&1; then
+        docker image rm "$IMAGE" >/dev/null
     fi
 }
 
@@ -39,22 +39,15 @@ main() {
         exit 1
     fi
 
-    IIDFILE="$(mktemp)"
     trap cleanup EXIT
     trap 'cleanup; exit 130' INT TERM
 
     echo "Building the pinned Firefox e2e image (extension included)..."
     # --no-cache: the image build runs make build and its checks, and a
     # cached layer is a check that did not run.
-    docker build --no-cache --iidfile "$IIDFILE" -t "$IMAGE" \
-        -f tests/e2e/firefox/Dockerfile .
+    docker build --no-cache -t "$IMAGE" -f tests/e2e/firefox/Dockerfile .
 
     echo "Running the Firefox e2e suite..."
-    # The image is run by ID, not by tag: where two clones of this repo run
-    # the suite at once, the other build can move the tag between this
-    # build and this run, and the suite would then silently test the other
-    # checkout.
-    #
     # --shm-size=1g: Firefox needs more than the default 64MB /dev/shm.
     # --network none: the suite stubs nothing, so this is what keeps the
     #   run offline and deterministic. The extension swallows its own
@@ -72,7 +65,7 @@ main() {
         --shm-size=1g \
         --network none \
         -e HOME=/tmp \
-        "$(cat "$IIDFILE")" \
+        "$IMAGE" \
         node tests/e2e/firefox/run.js dist/firefox
 
     # The install/uninstall/re-install property, against the packaged XPI
@@ -85,7 +78,7 @@ main() {
         --shm-size=1g \
         --network none \
         -e HOME=/tmp \
-        "$(cat "$IIDFILE")" \
+        "$IMAGE" \
         node tests/e2e/firefox/reinstall.js
 }