diff --git a/man/systemd.exec.xml b/man/systemd.exec.xml
index 7821b69cd0..53daff0f64 100644
--- a/man/systemd.exec.xml
+++ b/man/systemd.exec.xml
@@ -2012,6 +2012,12 @@ CapabilityBoundingSet=~CAP_B CAP_C</programlisting>
         <varname>$JOURNAL_STREAM</varname> is set at all as services might invoke external processes replacing their
         standard output or standard error output, without unsetting the environment variable.</para>
 
+        <para>If both standard output and standard error of the executed processes are connected to the journal via a
+        stream socket, this environment variable will contain information about the standard error stream, as that's
+        usually the preferred destination for log data. (Note that typically the same stream is used for both standard
+        output and standard error, hence very likely the environment variable contains device and inode information
+        matching both stream file descriptors.)</para>
+
         <para>This environment variable is primarily useful to allow services to optionally upgrade their used log
         protocol to the native journal protocol (using
         <citerefentry><refentrytitle>sd_journal_print</refentrytitle><manvolnum>3</manvolnum></citerefentry> and other
diff --git a/src/core/execute.c b/src/core/execute.c
index 9dcc02aaa4..e2d84feb0d 100644
--- a/src/core/execute.c
+++ b/src/core/execute.c
@@ -595,9 +595,13 @@ static int setup_output(
 
                         /* If we connected this fd to the journal via a stream, patch the device/inode into the passed
                          * parameters, but only then. This is useful so that we can set $JOURNAL_STREAM that permits
-                         * services to detect whether they are connected to the journal or not. */
+                         * services to detect whether they are connected to the journal or not.
+                         *
+                         * If both stdout and stderr are connected to a stream then let's make sure to store the data
+                         * about STDERR as that's usually the best way to do logging. */
 
-                        if (fstat(fileno, &st) >= 0) {
+                        if (fstat(fileno, &st) >= 0 &&
+                            (*journal_stream_ino == 0 || fileno == STDERR_FILENO)) {
                                 *journal_stream_dev = st.st_dev;
                                 *journal_stream_ino = st.st_ino;
                         }