Or does it exist mathematically? Is it really inconsistent with a common-sense, mathematics or known physical laws?
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.
It would be also very consistent with the idea that the total energy of the Universe is zero. Doesn't it imply that the total mass of the Universe is zero?
Can you please clarify the statements below that seem to contradict each other?
1) "all structures that exist mathematically exist also physically" -Max Tegmark.
2) 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.
3) 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.