A script causes a parent script to stop

Question

I have a third-party script that runs some complicated stuff.

I run everything from an ssh session (the machine is a headless VM running Ubuntu 20.04).

When I run it interactively, it runs normally.

$ ./third-party-script.sh one
<script output one>
$ ./third-party-script.sh two
<script output two>

When I run it from another script, the parent script stops when the third-party script executes for the second time.

$ cat my-script.sh
#!/bin/sh
./third-party-script.sh one
./third-party-script.sh two
$
$ ./my-script.sh
<script output one>
[1]+  Stopped                 ./my-script.sh
$

I can send my script to foreground and it continues normally

$ fg
<script output two>
$ 

What could be the reason of this behaviour and how can it be avoided? I cannot fix the third-party script or anything that it runs, I can only fix my script. (The original "my-script" is written in Python and uses os.system, I simplified it to a bare-bones shell script for this question).

I tried running the script with nohup but it didn't help.

Thanks to @ilkkachu, I have narrowed it down to the third-party script executing bash -ic some_command. So the minimal script that reproduces this in my environment is:

#!/bin/sh
bash -ic ls
bash -ic ls

sh -ic and bash -c both fix the problem. Unfortunately I cannot change the command. I need to wrap the third-party script somehow so that it stops doing this.

I tried bash -icx instead, but this doesn't narrow down the problem any further. The second bash execution stops before printing any command.

I tried to strace -f the whole thing and this is what comes in the end:

[pid 3805453] ioctl(255, TIOCGPGRP, [3805322]) = 0
[pid 3805453] rt_sigaction(SIGTTIN, {sa_handler=SIG_DFL, sa_mask=[], sa_flags=SA_RESTORER, sa_restorer=0x7f131f36a090}, {sa_handler=SIG_IGN, sa_mask=[], sa_flags=SA_RESTORER, sa_restorer=0x7f131f36a090}, 8) = 0
[pid 3805453] kill(0, SIGTTIN)          = 0
[pid 3805321] <... wait4 resumed>0x7ffcc2d45bfc, 0, NULL) = ? ERESTARTSYS (To be restarted if SA_RESTART is set)
[pid 3805453] --- SIGTTIN {si_signo=SIGTTIN, si_code=SI_USER, si_pid=3805453, si_uid=4437} ---
[pid 3805321] --- SIGTTIN {si_signo=SIGTTIN, si_code=SI_USER, si_pid=3805453, si_uid=4437} ---
[pid 3805453] --- stopped by SIGTTIN ---
[pid 3805321] --- stopped by SIGTTIN ---

3805453 is the second child bash running -ic ls and 3805321 is the top-level bash running my-script.sh.

It is the only line with kill(0, SIGTTIN) in the strace log. The first child bash doesn't do anything like that, only the second one does.