There are many tiles that are 2-anisohedral. The two transitivity classes required for these tiles will usually be the same size. (OK, OK, that's not precise. Look at ratios of classes in ever expanding disks or take the ratio in a block that tiles isohedrally.) But a tile has been discovered that requires one transitivity class to be twice the size of the other.

This tile will only tile in one way:

Note that we have two transitivity classes - colored red and yellow. Also note that the ratio is two yellow tiles for every one red tile. This may be seen by looking at a block of tiles which is isohedral.

There were two people involved in finding this tile. It was found by a computer search by Joseph Myers. He noticed that the tile had symmetry. I noticed that the ratio of the transitivity classes was 2:1. As his computer program was designed to find tilings, not to answer this specific question, I did a hand check to verify that there were no other ways to tile with this shape. Its classification is 5