VM inbound traffic only works with outbound traffic at same time

I've recently configured a VM on a low-end rental VPS service to try to improve load times for a web site I'm running. (The web site makes use of high-reliability technologies that require full access to a VM, so I can't just use cheap IIS site hosting.) I've got it all set up, and it's working well (and way faster than my cable modem :-) with one caveat, and it's an odd one.

The VPS doesn't seem capable of receiving inbound traffic unless it has just recently sent an outbound packet. So, if the server is just sitting there, and I'm connected to the "KVM" VNC connection, I can use the system, it's fully responsive, but it doesn't respond to pings or connection attempts on its Internet interface. As soon as I initiate some network traffic from the VPS outbound, such as ping requests, browsing a web page, downloading OS updates or what have you, then suddenly it starts getting & replying to pings. As soon as the outbound traffic drops off, the site drops off the Internet again.

The site is up and running right now only because I have an ongoing "ping" operation in a console window hitting another IP of mine. One outbound ping a second seems to keep the site pretty much accessible.

It seems very odd to me that the inbound traffic should be contingent on outbound traffic in this way. The VPS provider assures me it's nothing they're in charge of -- but then at the rates I'm paying I don't know if I expect them to understand every last detail of the service they're offering. :-)

Has anyone heard of this sort of behaviour before?

Here are all of the configuration details I know that I can think of, off the top of my head:

• Host OS: I believe Red Hat, but definitely a variant of Linux. The guest was preinstalled with Red Hat virtio drivers. I don't know any other details.
• Guest OS: Windows Server 2008 R2, fully patched
• Network drivers: Red Hat VirtIO Ethernet Adapter, 61.72.104.11000, built 2015-09-22