As far as I understand, if it exists, it must be far away from the "positive" matter because of repelling force, so it explains why there is no observations of such matter.
For one, negative mass would still be attracted to positive mass, but the positive mass would be repelled. This would lead to the negative mass "following" the positive mass.
Why is this? The force is a repulsive one. But we also have the fact that $\vec F=m\vec a$. Since one of the bodies has negative mass, it will be attracted.
If there was a body of negative mass less massive (talking about absolute values here) than our positive massed-universe, it would move much faster and eventually "catch up" with ours. If it had the same absolute value of mass, both would keep accelerating and it would never catch up. If it was larger, it would be left behind eventually.
The lack of any large quantities of negative mass in our universe excludes the case of a body of negative mass having caught up with us. The lack of any acceleration of the universe signifies that there isn't any large body of negative mass with absolute value less than or equal to the mass of the universe. So, if there is any large body of negative mass, it has larger mass than our universe and it is separated from our universe.
"all structures that exist mathematically exist also physically" -Max Tegmark.
This is a part of his MAthematical Universe hypothesis. Note that it's just a hypothesis, it isn't backed by much concrete evidence yet.
Despite being completely inconsistent with a common-sense approach and the expected behavior of "normal" matter, negative mass is completely mathematically consistent and introduces no violation of conservation of momentum or energy.
This is true--it lets one create energy out of thin air by introducing a body with negative mass to the system, but this doesn't conflict the principle of conservation of energy as the body has it's own energy become more negative.
Such matter would violate one or more energy conditions and show some strange properties, stemming from the ambiguity as to whether attraction should refer to force or the oppositely oriented acceleration for negative mass.
I'm not too sure about the energy conditions, I'm not strong in General Relativity. But the second half is just about some strangeness attached to it and some confusion regarding terminology (relating to what I explained near the top of this answer).
Finally, vacuum fluctuations (which exist) can have a net negative energy density, so these have "negative mass", in a way. But these aren't easy to harness with current technology.
I found a post here on physicsforums.com which has some useful links in the post by "pervect". One is to this article by physicist John Cramer, saying that each time a mass M passes through a wormhole mouth, "the entrance mouth has its mass increased by M, and the exit mouth has its mass reduced by an amount -M", and that this can eventually cause one of the mouths to have a net negative mass. Presumably light passing through a wormhole could have the same effect, based on mass-energy equivalence in general relativity. Pervect notes that in this context the "mass" being discussed is the ADM mass, and links to this post discussing the technical details of calculating the ADM mass of wormhole mouths, as well as some issues relating to quantum uncertainty in mass--I don't know enough general relativity to follow the technical details here, but the author seems to say that the mass of a mouth cannot actually become negative, perhaps for reasons relating to the "quantum inequalities" postulated to restrict negative energy that are discussed in this article. (maybe when John Cramer talked about the mass becoming negative he was giving the answer in "pure" general relativity without considering quantum physics?) Hopefully someone else who understands these topics better will weigh in, but I thought these links would be useful as pointers to research that would likely be relevant to answering your question.
Best Answer
Here is a flow chart of the forms of energy, with links .
Conservation of energy is one of the fundamental laws governing physical systems and is the only reason why one can talk of "negative energy"
here is a breakdown of the forms that **Conservation of energy ** takes
In almost all frames negative energy exists, in the sense of conservation of energy, for example between potential forms and kinetic forms. Look at the energy levels of the hydrogen atom, for example, as a consequence of conservation of energy. It is only in the special and general relativity where the concept of "negative energy" becomes problematic, when it ties up with the mass energy equivalence. Otherwise, negative energy means that, from conservation of energy, positive energy also exists in the system, because of the energy conservation law.
Thus the question has "unexpected" meaning only if it is asking whether negative masses exist, not in the formulation of your question, where the answer is "yes".
It is only negative masses that would behave the way you describe:
All these are results from the existence of negative mass, not negative energy, which exists in all classical systems.
The problem of whether antiparticles have also negative gravitational mass is under experimental exploration.
The experiments have a long way to go.