Tiling a rectangle with a single polyomino

Question

Is it true that if you can decompose a polyomino into multiple polyominoes none of which can tile a rectangle, then you can't tile a rectangle with the polyomino?

Answer 1

I can think of a simple example where this is not true, at least in the case where you decompose a polyomino into two polyominos with different sizes. Start with a $4\times5$ rectangle. Obviously you can tile a bigger rectangle (of compatible size) with such object. Now decompose it into two polyominos in the following way: take a cross (three rows, one square in the first row, three in the second, one in the third), and cut it out from the $4\times5$ rectangle, such that the top of the cross is in the middle of the long side. Neither the cross, nor the remaining figure can be used to tile any rectangle. I assume that you can even increase the size of the original polyomino and the size of the cross, so that the area of the cross is half of the area of the original polyomino, in case you want to decompose in equal size parts.