I have heard conflicting views on whether a EE site running FPC can also run Varnish, or whether to run Varnish, FPC must be disabled. Any help greatly appreciated.
If you take a look at the installation instructions for the most popular Varnish extension for Magento: https://github.com/nexcess/magento-turpentine/wiki/Installation
Clear the Config, Layout, and Block HTML caches and disable the Full Page Cache cache (if it is available and enabled) in System > Cache Management, then logout and log back in.
The reason for this is that they share the exact same purpose, with Varnish, you don't need FPC any more.
Varnish and FPC can be used great together. Serve Varnish pages for those who are simply window shopping and once they become a customer (login or add an item to their cart) you switch them over to FPC.
Varnish is great for static content and FPC is great at speeding up Magento with dynamic content. Bring them together and you'll have a very good customer experience. You'll need to finely tune your Varnish configuration or you may run into issues along the way.
FPC's and Varnish both cache at the page level. So never use a FPC with Varnish unless it's specifically meant to work with Varnish. You won't get better results by using both, but you will likely run into problems with customer specific content in it. You should choose one or the other.
Should you choose the FPC route, we have a Full Page Cache extension that we have been developing and supporting for a number of years. Brim's Full Page Cache, http://ecommerce.brimllc.com/full-page-cache-magento.html.
We follow the best Magento coding practices for our extension, and no core Magento files ever need to be touched. We also play as nice as possible with other extensions. We have full support for hole punching, multiple stores, currencies, languages, etc. On top of it all, we have a 15 day refund policy and great support should you need it.