Description of problem:

I don't know if the issue is with bash or sshd, but I believe it's more sshd since the same can be reproduced with zsh but not ksh.
When having a HEREDOC processed through ssh, it appears that any stdin consumer will "eat" the next lines in HEREDOC instead of reading on standard input, as shown in the example below:
-------- 8< ---------------- 8< ---------------- 8< ---------------- 8< --------
$ ssh -t -t localhost << EOF
tty
cat
echo HELLO
EOF

tty
cat
echo HELLO
[root@vm-rhel8 ~]# tty
/dev/pts/1
[root@vm-rhel8 ~]# cat
echo HELLO
-------- 8< ---------------- 8< ---------------- 8< ---------------- 8< --------

On RHEL7, we have the "cat" command wait for input, whereas on RHEL8 and RHEL9, we have the "cat" command read next line ("echo HELLO") then wait for input.
The same is reproducible with zsh, but not ksh:
-------- 8< ---------------- 8< ---------------- 8< ---------------- 8< --------
$ ssh -t -t localhost ksh << EOF
tty
cat
echo HELLO
EOF
tty
cat
echo HELLO
# 
/dev/pts/1
#
---> this is "cat" execution waiting on standard input. "echo HELLO" was not processed yet.
-------- 8< ---------------- 8< ---------------- 8< ---------------- 8< --------

Version-Release number of selected component (if applicable):

openssh-server-8.0p1-13.el8.x86_64

How reproducible:

Always

Steps to Reproduce: see above

Actual results:

"echo HELLO" being processed by "cat"

Expected results:

"cat" just waiting for input

Additional info:

There is a real issue behind this. As soon as a program "eats" stdin, such as "systemctl" commands, weird behaviour happens, as shown below:

- On RHEL7, no issue

Command:
-------- 8< ---------------- 8< ---------------- 8< ---------------- 8< --------
# ssh -t -t vm-rhel7 << EOF
systemctl status sshd
echo "RC $?"
exit
EOF

systemctl status sshd
echo "RC 0"
exit
Last login: Thu Jun 16 09:39:32 2022 from localhost
[root@vm-rhel7 ~]# systemctl status sshd
● sshd.service - OpenSSH server daemon
   Loaded: loaded (/usr/lib/systemd/system/sshd.service; enabled; vendor preset: enabled)
   Active: active (running) since Thu 2022-06-16 08:55:23 CEST; 46min ago
     Docs: man:sshd(8)
           man:sshd_config(5)
 Main PID: 1044 (sshd)
    Tasks: 1
   Memory: 5.7M
   CGroup: /system.slice/sshd.service
           └─1044 /usr/sbin/sshd -D

Jun 16 09:33:08 vm-rhel7 sshd[4279]: Accepted publickey for root from ::1 p...E4
Jun 16 09:33:19 vm-rhel7 sshd[4366]: Accepted publickey for root from ::1 p...E4
Jun 16 09:33:53 vm-rhel7 sshd[4473]: Accepted publickey for root from ::1 p...E4
Jun 16 09:34:19 vm-rhel7 sshd[4602]: Accepted publickey for root from ::1 p...E4
Jun 16 09:35:19 vm-rhel7 sshd[4767]: Accepted publickey for root from ::1 p...E4
Jun 16 09:35:30 vm-rhel7 sshd[4853]: Accepted publickey for root from ::1 p...E4
Jun 16 09:36:14 vm-rhel7 sshd[4987]: Accepted publickey for root from ::1 p...E4
Jun 16 09:39:08 vm-rhel7 sshd[5363]: Accepted publickey for root from ::1 p...E4
Jun 16 09:39:32 vm-rhel7 sshd[5485]: Accepted publickey for root from ::1 p...E4
Jun 16 09:41:33 vm-rhel7 sshd[5699]: Accepted publickey for root from 192.1...E4
Hint: Some lines were ellipsized, use -l to show in full.
[root@vm-rhel7 ~]# echo "RC 0"
RC 0
[root@vm-rhel7 ~]# exit
logout
Connection to vm-rhel7 closed.
-------- 8< ---------------- 8< ---------------- 8< ---------------- 8< --------

- On RHEL8, breaks due to pager (started by "systemctl status") eating "echo "RC $?"" line
-------- 8< ---------------- 8< ---------------- 8< ---------------- 8< --------
# ssh -t -t vm-rhel8 << EOF
systemctl status sshd
echo "RC $?"
exit
EOF

systemctl status sshd
echo "RC 0"
exit
[root@vm-rhel8 ~]# systemctl status sshd
● sshd.service - OpenSSH server daemon
   Loaded: loaded (/usr/lib/systemd/system/sshd.service; enabled; vendor preset>
   Active: active (running) since Thu 2022-06-16 08:53:15 CEST; 48min ago
     Docs: man:sshd(8)
           man:sshd_config(5)
 Main PID: 896 (sshd)
    Tasks: 1 (limit: 11377)
   Memory: 9.5M
   CGroup: /system.slice/sshd.service
           └─896 /usr/sbin/sshd -D -oCiphers=aes256-gcm,chacha20-po>

Jun 16 09:36:25 vm-rhel8 sshd[2832]: pam_unix(sshd:session): session opened for>
Jun 16 09:36:41 vm-rhel8 sshd[2895]: Accepted publickey for root from 192.168.1>
Jun 16 09:36:41 vm-rhel8 sshd[2895]: pam_unix(sshd:session): session opened for>
Jun 16 09:39:19 vm-rhel8 sshd[2958]: Accepted publickey for root from 192.168.1>
Jun 16 09:39:19 vm-rhel8 sshd[2958]: pam_unix(sshd:session): session opened for>
Jun 16 09:39:19 vm-rhel8 sshd[2958]: pam_unix(sshd:session): session closed for>
Jun 16 09:39:39 vm-rhel8 sshd[3026]: Accepted publickey for root from 192.168.1>
Jun 16 09:39:39 vm-rhel8 sshd[3026]: pam_unix(sshd:session): session opened for>
Jun 16 09:41:55 vm-rhel8 sshd[3078]: Accepted publickey for root from 192.168.1>
Jun 16 09:41:55 vm-rhel8 sshd[3078]: pam_unix(sshd:session): session opened for>
...skipping...

                   SUMMARY OF LESS COMMANDS

      Commands marked with * may be preceded by a number, N.
      Notes in parentheses indicate the behavior if N is given.
      A key preceded by a caret indicates the Ctrl key; thus ^K is ctrl-K.
...
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