Run "git" on remote server with drush alias
This works just fine for me on both the 6.x and master branches of Drush, and on the 6.2.0 release.  So, your first step would be to try with the latest versions, and see if it works.  If that doesn't fix your problem, try running your command again using the -v flag to get more verbose output.  You will see something like this:

$ drush -v @mysite.prod git-status
Loaded alias @server.xxxxxxxxxxx from file                              [notice]
/home/ga/local/config/drush-aliases/server.aliases.drushrc.php
Loaded alias @mysite.prod from file                                     [notice]
/home/ga/local/config/drush-aliases/mysite.aliases.drushrc.php
Include /home/ga/local/drupal/drush_extensions                          [notice]
Include /home/ga/.drush                                                 [notice]
Calling proc_open(ssh -o PasswordAuthentication=no webuser@myserver.com 'cd /srv/www/myserver.com/htdocs &amp;&amp; git status');
# On branch master
# Untracked files:
#   (use "git add &lt;file&gt;..." to include in what will be committed)
#
#   ../xxxxx.txt
nothing added to commit but untracked files present (use "git add" to track)
Command dispatch complete                                               [notice]


Try running the command inside of "proc_open(...)" from the shell, and see if it behaves the same way.  Adjust the ssh options until it works right.  Then, set $options['ssh-options'] to adjust.

NOTE:

If you want to do something like this without defining a shell alias, the following should also work:

drush @mysite.prod ssh git status