Who should be first author? Theory person or Experiment person? The situation:

Two PhD students named 'Theory' and 'Experiment' are working on a project. Both are experimentalists by training but, from experiences working on a separate project, Theory comes up with an idea for a new experiment. Positive results from this experiment would have a small but significant impact on their field. However, Theory does not have expertise on the particular instrument required to conduct the experiment so, with the blessing of their adviser, Theory enlists Experiment who is an expert at using the instrument. Theory guides the experiment but the execution is all done by Experiment.

Happily, the results are positive! 

The question is, given that Theory and Experiment agree to share first authorship, who should be listed first? 

Edit: after writing this I found this thread
Had the idea vs. did the work: Who should be the lead author? which is very similar to this question.

There's also an interesting discussion of the issue in this blog: https://funkdoctorx.wordpress.com/2012/02/27/co-first-author-and-cvs/
Shared first authors are listed alphabethically, at least that's what makes sense, otherwise they are not shared first authors!! I'd go with that no doubt.

From your comments I would say that your are the Experiment person, as you seem to be diminishing the role of the Theory person (he/she simply did so and so...).

From your description, the Theory person came up with the idea and guided the experiments, while the Experiment person simply run the experiments. As a general principle, I would tend to side with the Theory person (Peter's answer here seems to corroborate my view). I believe it also matters who is going to write most of the paper, which you do not mention. 

However, as we are all a bunch of strangers on the internet, there's no actual way of knowing which is the case (the idea may be too simple or the experiment may be very complex, and so on...). The best thing to do would be to talk to your supervisor.

Most importantly, don't cling on unimportant details, as you already have a shared first-author status. Move on. This is the kind of meaningless thing that can ruin a scientific collaboration for good.