  Let $\mathcal{T}_n$ be the set of ribbon $L$-shaped $n$-ominoes for some
$n\ge 4$ even, and let $\mathcal{T}_n^+$ be $\mathcal{T}_n$ with an extra
$2\times 2$ square. We investigate signed tilings of rectangles by
$\mathcal{T}_n$ and $\mathcal{T}_n^+$. We show that a rectangle has a signed
tiling by $\mathcal{T}_n$ if and only if both sides of the rectangle are even
and one of them is divisible by $n$, or if one of the sides is odd and the
other side is divisible by $n\left (\frac{n}{2}-2\right ).$ We also show that a
rectangle has a signed tiling by $\mathcal{T}_n^+, n\ge 6$ even, if and only if
both sides of the rectangle are even, or if one of the sides is odd and the
other side is divisible by $n\left (\frac{n}{2}-2\right ).$ Our proofs are
based on the exhibition of explicit Gr\"obner bases for the ideals generated by
polynomials associated to the tiling sets. In particular, we show that some of
the regular tiling results in \emph{ V.~Nitica, Every tiling of the first
quadrant by ribbon $L$ $n$-ominoes follows the rectangular pattern. Open
Journal of Discrete Mathematics, {\em 5}, (2015) 11--25,} cannot be obtained
from coloring invariants.
