  We construct a family of planar self-affine carpets with overlaps using lower
triangular matrices in a way that generalizes the original Gatzouras--Lalley
carpets defined by diagonal matrices. Assuming the rectangular open set
condition, Bara\'nski proved for this construction that for typical parameters,
which can be explicitly checked, the inequalities between the Hausdorff, box
and affinity dimension of the attractor are strict. We generalize this result
to overlapping constructions, where we allow complete columns to be shifted
along the horizontal axis or allow parallelograms to overlap within a column in
a transversal way. Our main result is to show sufficient conditions under which
these overlaps do not cause the drop of the dimension of the attractor. Several
examples are provided to illustrate the results, including a self-affine
smiley, a family of self-affine continuous curves, examples with overlaps and
an application of our results to some three-dimensional systems.
