This is already implemented. See the following config option on the master-1.2.x branch - not yet ported to master:

/***************
 * MantisTouch *
 ***************/

/**
 * The MantisTouch URL to direct to.  The %s will be replaced by the contents of $g_path.
 * A blank value will disable redirection.  The %s is not required when redirecting to
 * MantisTouch instances that point to a single MantisBT instance and hence have a hard-coded URL.
 *
 * Following are three examples:
 * - 'http://mantisbt.mobi?url=%s'
 * - 'http://MyOwnMantisTouch.com/'
 * - ''
 */
$g_mantistouch_url = '';

This is not what I mean.
I asked for an additional option.

Example 1
$g_mantistouch_url = 'http://mantisbt.mobi?url=%s'
$auto_redirect_to_mobile = ON;
If a mobile browser is detected, redirect to mobile (current implementation)

Example 2
$g_mantistouch_url = 'http://mantisbt.mobi?url=%s'
$auto_redirect_to_mobile = OFF;
If a mobile browser is detected, ask the user if he wants to use the mobile pages or the "classic" pages.

IMO this is needed as long as the mobile solution does not offer all MantisBT functionality.

With current implementation you are not able to access www.mantisbt.org/bugs with a mobile device and have a look for example at the summary page, your account settings, ...

I don't think we should add such option. What we should do is provide a way for the user to switch from mobile to full version on demand. Once the user selects one of the two options, it should stick until the user switches again.

What we should do is provide a way for the user to switch from mobile to full version on demand
Fine for me, but might be more difficult to implement.