For a convex tile every point P of the straight line segment between two arbitrary points A and B of the tile also has to belong to the tile.
We consider 3 cases.
Case 1: Two tiles completely surround the other two tiles.
There are two points which the blue and the red tile have in common together with a third tile. As these points both belong to the red and the blue tile, all points along the red line have also belong to the red and the blue tile, if the tiles would be convex. But then there is no space left for the surrounded tiles (yellow and green).
Case 2: Three tiles completely surround one tile and the tetrad has no hole.
(A mathematician would not mention this case, because it follows from case 3.)
There are three points that belong to three tiles. These points all belong to the inner tile (yellow). If the tiles would be convex, then each straigt line between two red points would have to belong to the inner tile and one of the surrounding tiles. Therefore the inner tile would be a triangle. But triangles cannot form a tetrad.
Case 3: Three tiles completely surround one tile and the tetrad has holes.
Consider three congruent tiles that are convex and surround an empty space. There are three points A, B, and C which belong to two tiles and addtionally touch the empty space. Each of the straigt lines AB, BC, and CA is a subset of a tile. Therefore the interior of the triangle ABC is the maximal possible empty space. Say s is the maximum length of the triangle sides. The missing inner tile also has to have an edge with length s, because it is congruent to the other tiles. The only way to put the missing tile in the empty space is, that it has its side with length s in common with another tile (the green tile in the second figure).
The maximum area of the green tile is a trapezoid, as it is congruent to the yellow tile. But the green tile has only one point in common with the red tile (and with the blue tile). It would be necessary that the red tile touches a short part of the edge on the right of point B (see last figure). Say this part ends at Point D. But then the straight line segment AD is not a subset of the red tile and thus the red tile is not convex. The tiles cannot be convex.