diff --git a/README.md b/README.md
index 8e64837..9c6ce5e 100644
--- a/README.md
+++ b/README.md
@@ -347,6 +347,14 @@ 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, so two clones
+can run the same suite at once without one build taking the other's image away.
+The run removes that image, and its container if one is still running, after the
+suite passes or fails, on Ctrl-C, on `kill`, and when the terminal or SSH
+session it runs in closes. A run killed outright (`kill -9`) cannot clean up:
+its image stays under its tag until removed by hand with `docker image rm`.
+
 ### 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..2763bb6 100755
--- a/script/test-e2e
+++ b/script/test-e2e
@@ -22,13 +22,29 @@ 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.
+SUFFIX="$(od -An -N6 -tx1 /dev/urandom | tr -d ' \n')"
+NAME="$("$SCRIPT_DIR/projectname")-e2e-chrome"
+IMAGE="$NAME:$SUFFIX"
+# Both docker runs below use this container name: docker run --rm returns
+# only once its container is removed, so the second never meets the first.
+CONTAINER="$NAME-$SUFFIX"
 
 cleanup() {
-    if [ -n "$IIDFILE" ]; then
-        rm -f "$IIDFILE"
+    # After a third Ctrl-C docker's client exits whether or not the
+    # container has stopped, and a container still running keeps the image
+    # in use.
+    if docker container inspect "$CONTAINER" >/dev/null 2>&1; then
+        docker container rm -f "$CONTAINER" >/dev/null
+    fi
+    # 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 +56,16 @@ main() {
         exit 1
     fi
 
-    IIDFILE="$(mktemp)"
     trap cleanup EXIT
-    trap 'cleanup; exit 130' INT TERM
+    # HUP: the terminal or SSH session running the suite closed.
+    trap 'cleanup; exit 130' HUP 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
@@ -75,11 +85,12 @@ main() {
     #   probe. The image is pinned by digest, so this can only ever bite
     #   on a deliberate bump.
     docker run --rm \
+        --name "$CONTAINER" \
         --ipc=host \
         -e HOME=/tmp \
         -e PW_EXPERIMENTAL_SERVICE_WORKER_NETWORK_EVENTS=1 \
         -e "E2E_TRACE_NETWORK=${E2E_TRACE_NETWORK:-0}" \
-        "$(cat "$IIDFILE")" \
+        "$IMAGE" \
         node tests/e2e/run.js
 
     # Where chrome.storage.local lives, and what moves it: two unpacked loads
@@ -93,9 +104,10 @@ main() {
     # --host-resolver-rules=MAP * ~NOTFOUND so nothing it does can leave.
     echo "Running the extension-id and storage-partition observations..."
     docker run --rm \
+        --name "$CONTAINER" \
         --ipc=host \
         -e HOME=/tmp \
-        "$(cat "$IIDFILE")" \
+        "$IMAGE" \
         node tests/e2e/storagePartition.js
 }
 
diff --git a/script/test-e2e-firefox b/script/test-e2e-firefox
index 892e7e1..884a710 100755
--- a/script/test-e2e-firefox
+++ b/script/test-e2e-firefox
@@ -21,13 +21,23 @@ set -eu
 SCRIPT_DIR="$(cd "$(dirname "$0")" && pwd -P)"
 ROOT="$(cd "$SCRIPT_DIR/.." && pwd -P)"
 
-IMAGE="$("$SCRIPT_DIR/projectname")-e2e-firefox"
-
-IIDFILE=""
+# A tag and a container name of this run's own, as in script/test-e2e,
+# which says why.
+SUFFIX="$(od -An -N6 -tx1 /dev/urandom | tr -d ' \n')"
+NAME="$("$SCRIPT_DIR/projectname")-e2e-firefox"
+IMAGE="$NAME:$SUFFIX"
+CONTAINER="$NAME-$SUFFIX"
 
 cleanup() {
-    if [ -n "$IIDFILE" ]; then
-        rm -f "$IIDFILE"
+    # After a third Ctrl-C docker's client exits whether or not the
+    # container has stopped, and a container still running keeps the image
+    # in use.
+    if docker container inspect "$CONTAINER" >/dev/null 2>&1; then
+        docker container rm -f "$CONTAINER" >/dev/null
+    fi
+    # 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 +49,16 @@ main() {
         exit 1
     fi
 
-    IIDFILE="$(mktemp)"
     trap cleanup EXIT
-    trap 'cleanup; exit 130' INT TERM
+    # HUP: the terminal or SSH session running the suite closed.
+    trap 'cleanup; exit 130' HUP 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
@@ -69,10 +73,11 @@ main() {
     # "CanCreateUserNamespace() clone() failure: EPERM" on startup here;
     # it is cosmetic and headless Firefox runs fine without it.
     docker run --rm \
+        --name "$CONTAINER" \
         --shm-size=1g \
         --network none \
         -e HOME=/tmp \
-        "$(cat "$IIDFILE")" \
+        "$IMAGE" \
         node tests/e2e/firefox/run.js dist/firefox
 
     # The install/uninstall/re-install property, against the packaged XPI
@@ -82,10 +87,11 @@ main() {
     # and the suite above shares one session throughout.
     echo "Running the Firefox re-install suite against the packaged XPI..."
     docker run --rm \
+        --name "$CONTAINER" \
         --shm-size=1g \
         --network none \
         -e HOME=/tmp \
-        "$(cat "$IIDFILE")" \
+        "$IMAGE" \
         node tests/e2e/firefox/reinstall.js
 }