I haven't written down an example, but I believe the following would be possible:

(1) both triangles must be scalene (all sides, and thus all angles of the triangle must be different)

(2) the triangles are similar (thus they have the same angles)

(3) the side opposite the smallest angle of the first triangle is congruent to the side opposite the middle angle of the second triangle.

(4) the side opposite the middle angle of the first triangle is congruent to the side opposite the largest angle of the second triangle.