The only issue that I know I will never be able to get over is her being physical with another man......But, I do want to be able to be okay with her having another relationship without the physical aspect.
Our situations are somewhat different - a few months back, my wife of several years asked me to open our marriage to poly so she could explore her resurgent feelings for an old college bf, while your gf told you that she was poly upfront (although not physically intimate with her long distance bf). But I do very much relate with your feelings about her being physically intimate with another man. Just a few years ago, I would almost certainly have made a similar statement - as my cultural conditioning in this area was quite strong. And even though I was able to make the transition to poly and eventually become comfortable with the idea of my wife being intimate with another man, for me - this was still the most difficult aspect of transitioning to poly.
I might first suggest that it might not be a completely practical idea to believe that a poly relationship could be maintained indefinitely without sex - not impossible, especially if one partner is asexual, or if there are other unusual circumstances such as health issues, etc. However, most healthy adult poly relationships do seem to eventually include sex. It just seems to be a natural progression in most adult romantic relationships. If this is truly a hard limit for you, and your gf is committed to poly - it is a difficult situation. I do believe that I too often hear splitting up suggested as the only solution on this forum (mono guy, at heart, that I am) - but you really might have to take a hard, honest look at your relationship.
And, I will ask - are you absolutely certain that this is a hard limit for you?This is the question I faced. In my case, I was willing to see that my feelings were a result of my cultural conditioning (deep south, evangelical) - and that if I had been raised in a society where poly was the norm, I would almost certainly feel differently. So, it was a learned - and not inherent - belief/feeling. Given that, I made it a point to familiarize myself with poly - participating in this forum, reading on books on poly, etc. - an intentional "de-conditioning" - just to be fair to all points of view. And, I did a lot of processing - ultimately asking if sex was really that sacred - after all, we had both had numerous other sex partners prior to our marriage. And, she told me from the start that I was also welcome to have another partner as well (easier said than done undoubtedly). At the end of the day, I was able to achieve a paradigm shift that allowed me to be ok with the idea. And, of course, you may not be able to relate to this at all.
For some men, the motivation of having additional sexual relationships might be enough to allow their ladies the same pleasure. But it does not seem fair - unless all agree - that the man should be able to have sex with other ladies, but the women should not be able to do the same with the gentlemen. Or so it seems to me.
Anyway - just a couple of thoughts from another guy that's been there. Al