Okay, so you open a web site over SSL, and it has something (ActiveX, Java,
etc.) running as a plug-in on that page which sets up a TS session to your
office? I've seen a few of those before, though I don't have a convenient
one to test with.
That sounds like IE7 is not permitting the plugin to maintain its connection
and continue working when it doesn't have focus.... I don't know of any IE7
setting that would do that offhand, but it's possible. One thing you could
try, if that is the issue, is running IE7 with Protected Mode turned off
(under Internet Options, Security tab). I don't recommend leaving it this
way if you can avoid it, but it's something to check if a web plug-in can't
work properly.
If the latter fixes it, particularly, then your company should follow up
with the makers of that client to make it play nicely in IE7 protected mode.