Nepenthes_AK
I'm not sure if you were looking for advice, or if it is still relevant anymore, but I'll throw in my two cents anyway. Lacking any proper context for the scenario I may be way off, but on that off chance it helps...
I think the most important first step to deciding how to proceed in your relationship, or even your friendship, is to identify what you want out of it. It seems so straightforward and simple, but can turn out to be quite complicated. Many people enter relationships without any kind of real goal, and it placed undue stress on both partners. While that purpose should remain malleable, and you can always update and change it, entering a relationship, or making a decision regarding one, without understanding your own desires can cause a lot of hardship for both sides.
At this point I think you need to ask yourself: What did I want out of this relationship? What do I want out of a relationship now? Will I be able to get that out of a relationship with this person?
I know nothing of your situation, but I do know that failing to answer these key questions leads to a lot of hardships and eventual breakups in a huge number of cases. It is often taken as an axiom that the other people you interact with want the exact same thing as you, and that is rarely the case. Without understanding what the other person truly wants and, more importantly, what you want, you can spend a lot of time tugging against one another instead of working together to build that perfect relationship where you would both be happy.
I know that for many identifying their own reasons can sometimes be difficult. If it helps, some of the more common ones I can think of include:
Finding a soul mate
Finding an activity partner
Sexual satisfaction (Sorry, I do not know your age)
Intellectual stimulation
Popularity via proxy
To meet social norms (ie: because you think it's what you're supposed to do)
etc...
The first four, for example, are valid reasons to be in a relationship, and understanding how well that relationship is fulfilling that purpose helps you determine if the person you are with is the right one. Since I was around 13 years old my primary goal was the first one listed. As such, my first two relationships with girls who wanted only short term relationships, possibly for the latter reasons, ended very quickly. In retrospect I am glad that I cut them off before there was too much attachment. It wasn't the easiest decision, but I maintain that it was the correct one. On the other hand, if you were in this relationship for one of the latter two reasons, it might be a good time to re-evaluate your position as neither is really a valid rationale for being intimate (and, even so, number six is extremely common).
It sounds as though you do not have complete trust in this person, and that they may have had incompatible qualities with you that made this breakup more of a relief than a tragedy. I am going to editorialize a bit here and say that without trust and respect, you will not have a healthy relationship. No matter what else you do, lack of trust will only breed jealousy, and lack of respect will never allow stability.
In anything you do in life, business, friendship, education, and especially relationships, one of the most important aspects is going to be clear concise communication. I have become all too aware of this myself over the past year or two, as my job commonly entails spending 45 minutes working on a situation that could have been resolved in under 10 minutes if it was communicated properly the first time. But I digress...
So it is in relationships too. The tired cliche of arguments erupting regularly because the woman expects that the man prioritizes long-term emotional connection while his main concern is physical satisfaction is a perfect example. If both communicated their goals they could hopefully come to a compromise, respecting one another's feelings and opinions, and further satisfying one another's desires. It makes a more stable and happy relationship for both. In a worse case scenario the relationship may fall apart, but that only allows both to find more compatible partners.
My advice, if she is still interested in getting back together, is to sit down and talk. What I mean by this is leaving as many emotions as possible at the door, losing any social inhibitions that may obscure the truth, and clearly communicate exactly what you feel and what you want out of your relationship. Then listen as she does the same. Just as importantly, determine exactly what caused the breakup, and any prior arguments, and plan a way to prevent a re-occurrence. I think you will find that in many cases it was nothing more than a miscommunication, and that can be easily rectified for the future. But keep in mind that this will only work if she is willing to be open and honest, and with the aforementioned goal, as well.
If you can do this, and begin a clear, honest, and open dialogue, it will become abundantly clear whether she is being honest or trying to play the situation to keep you together. Be aware of if she is doing the latter, but also be aware that not everyone, male or female, has the motives attributed in the past few posts. People are individuals, and if you give respect to no one because of what some might do, you do a disservice only to yourself.
Even if you do not get back together, doing this may help you gain closure. At the very least that closure would allow you to avoid any awkwardness you're feeling now.
Remember those tears may have indicated she sincerely wants to stay with you, even though doing so may not be a good idea. Sometimes two people simply aren't compatible, without fault on either side. There is not always blame, so don't feel the need to assign it if it is not there. I hope you can still maintain a friendship with her when this is all over.
Good luck; I hope things work out for you.
PS ~ My apologies for any spelling mistakes or vagueness; I am short on time and cannot re-read before posting this time...