How do I synchronize Thunderbird tags with Gmail labels?

"It seems more people wanted this feature, but you should realize Thunderbird's folders are Gmail labels. Therefore Thunderbird's tags are not the same as the labels, in other words: there is no Gmail counterpart.
So as of yet I don't think this feature is available. I will check to see if I can find a way to filter your inbox for tags (moving them into folders).

From the Thunderbird Knowledge Base on filters

Filtering based on Tags
Thunderbird doesn't have explicit
support for filtering by tags but you
can select Custom Headers from the
leftmost list box while editing the
filter rule and add support for
X-Mozilla-Keys, and then test whether
it ""contains"" that tag. Thunderbird
uses that header to store tags. For
example, X-Mozilla-Keys: forum fyi
indicates the message has the forum
and fyi tags.
IMAP accounts store tags
using message attributes instead, so
its not possible to do that with
messages stored in remote folders. If
you copy the message to a local folder
it currently doesn't add a
X-Mozilla-Keys header with the tags,
though it will add a empty
X-Mozilla-Keys header when you compact
it

Here's a link to the Knowledge Base on tags

Thunderbird has limited support for
tags in message filters. It just
supports set a tag as a action.
However, if you add a custom header
for the tag, or specify it using
""Customize"" in the leftmost list box
you can test for tags. Since you can
have multiple tags for a message its
recommended you test whether
X-Mozilla-Keys: ""contains"" a tag
rather than using ""is"".
View -> Sort by -> Tags can sort the
messages in a folder based on the tag.
View -> Messages -> Tags can be used
to display just the messages with the
selected tag. The View list box has a
similar menu command but it is not
visible in the toolbar in version 2.x.
Right click on the toolbar, drag and
drop the View control to the toolbar,
and press OK to add it. You can also
create a custom view using View ->
Messages -> Customize that displays
just the messages with the desired
tags."

"I was also looking for a way to syncronize gmail labels and thunderbird tags.
I (sortof) found a way ...
The gmail label to a message gets applied from my imap gmail filter. And I create an like-named tag in Thunderbird.

In gmail in a browser: Settings >> Filters >> Create a new Filter:
eg: Matches: subject:([nbusers])
Do this: Skip Inbox, Apply label ""[Gmail]/nbusers""
In Thunderbird:

Create a new Tag with the same name as the gmail label.
Tools >> Message Filters >> Select from ""Filters for:"" dropdown box >> New...
Add your criteria (eg: Subject contains [nbusers])
Add your actions (eg: ""Tag Message"", and select the new tag you created.)
To apply to existing messages, in the ""Message Filters"" window,
Highlight your new filter,
Select the gmail folder in the ""Run selected filter(s) on"" dropdown box at the bottom of the window
(eg: [gmail/nbusers]), and
Click the ""Run Now"" button.

I said ""sortof"" above, because it's not really synchronized in an automatic sense. You've got to set it up in both tools.

An update: I tried the mozilla-x-keys method for my thunderbird filter. It didn't work.
Also in my above example, my gmail filter has ""skip inbox"". This negates the thunderbird filter. It causes the thunderbird filter to not see the incoming message and so it doesn't act on it. It works if you take off the ""skip inbox"" criteria."