What is the strongest evidence that anti-matter exists?

Anti-matter is produced all the time in the world's particle accelerators. It is also produced natrually during air shower cascades caused by cosmic ray particles and gamma-rays interacting in the atmosphere and when these same particles interact in particle detectors.

One of the simplest methods to show it exists is used by the cosmic ray detector on the Pamela satellite. In this detector, there is a target that the cosmic rays (and gamma rays) interact with. Then the particles pass through a strong magnetic field. Since the path of moving charged particles curves in the presense of a magnetic field, they can see where the particle entered and where it hit the final target which is a calorimeter that measures the energy.

Electrons come into the detector and curve one direction. Positrons, the positive anti-particle to the electron, enter the detector and curve the other direction. They have the same mass and energy but opposite charges and this is easily measured showing that positrons exist.

The same principle is used to detect them in the particle accelerators around the world.

We also make use of anti-matter daily in Medical physics in the Positron Emission Tomography (PET)

When we look towards the center of our Milky Way galaxy (in the constellation Sagittarius), we can detect gamma-ray emission of energy 511 keV (kilo electron volts) which represents the energy of annihilation of an electron and a positron. A positron the anti-matter equivalent of an electron, and has a positive charge instead of a negative charge. Electrons and positrons each have rest mass of 511 keV (from Einstein's mass-energy equivalence E = mc^2) and when they meet, they mutually annihilate, giving off two of these 511 keV gamma rays.

So the fact that we observe these 511 keV gamma rays is proof of the existence of anti-matter in the universe at large, and not just in our accelerators.