Double tiling congruent triangles with little else in common

When you really want to tile more than one layer
but triple tiling is just too much of a good thing,
surely the happy medium is double tiling.

• How may a mosaic of more than 900 sections be double tiled
  with congruent triangles along the 6 guidelines listed below?

Here are two examples of double tiling with congruent triangles.
The first example demonstrates most guidelines of this puzzle
while the second follows the toughest guidelines as well.

In the first example, eight overlapping
26.6°- 63.4°- 90°
triangles double tile a square mosaic of
15 sections where:

1. “Double tiling” is taken to mean that
   every section of a mosaic is completely covered
   by portions of exactly two tiles
   and that all tiles lay completely within that mosaic.

2. The tiles are
   congruent
   triangles.

3. Each tile is uniquely oriented.

4. The mosaic is edge-contiguous in that all sections may be
   visited along a single unbroken path that stays within the mosaic
   while crossing tile edges from section to section
   without touching any vertex.

In the second example, four overlapping congruent
30°- 60°- 90°
triangles double tile a triangular mosaic of
4 sections where, moreover:

5. Every angle is a whole number of degrees.

6. No distinct lines are parallel.
   (Parallel tile edges may, however, lie along a single continuous line.)

Bounty challenges, achievability unknown

• Double tile a mosaic other than the second example above
  that follows all 6 guidelines and has no holes.

• Double tile a mosaic that follows all 6 guidelines
  and whose outline is not
  bilaterally symmetric.

(All interesting double tilings,
including those with fewer than 901 sections
and/or those that disregard some of the above guidelines,
deserve votes of approval.)