Answer: tracks left by Strongyloides stercoralis larvae. In this "culture" technique, stool is typically placed on a beef extract agar plate and incubated at room temperature. If Strongyloides stercoralis larvae are present, they will move out of the stool specimen, dragging bacteria with them. What we're actually seeing in this case the bacteria growing in the wake of the moving larva. These "tracks" can be seen using a dissecting microscope, and are typically present 24-48 hours after initial plating. Using a 4x objection, it is also possible to see "furrows" left in the agar by migrating larvae (without associated bacterial colonies) and the worms themselves.
Since this was a sputum sample, we were concerned that there was not enough bacteria present to allow the larva to create tracks; therefore, we made a 2:1 concentration of sample to Escherichia coli mixture and plated this on the agar.
A few interesting facts about Strongyloides stercoralis culture:
1. This technique was first described by Arakaki et al (1988).
2. In general, it increases the yield of positive cases by 50% or more compared to the formalin-ether concentration method.
3. Furrows and bacterial tracks are suggestion of a migrating larva, but definitive diagnosis is best accomplished by identification of the worm itself.
3. Other nematode larvae, such as free-living larvae and hookworm larvae, can be seen with this technique, so microscopic examination of the worm is important for differentiation. Of note, you would only expect to find these 2 larvae in stool samples that have sat at room temperature for an extended period (e.g. in hookworm infection, since this allows the eggs to hatch into L1 larvae), or if the sample was contaminated by soil that may contain free living nematodes.
4. Finally, as suggested by cmassey, the tracks resemble an aerial photo of the Amazon river flowing into the ocean!
Thanks to everyone who wrote in with comments.