?Presumably you've ruled out sample (or slide) contamination by dust, etc.? If really within the sediment samples, seems this must be biologic material (wild guess--filamentous algae preserved in anoxic lake bottom sediment environment--just a wild guess, I do not do work in microscopic examination of sediments!)
Yeah, I thought dust at first as well. I went down the forensics rabbit hole for bit with different fibers, but they are much larger than these things. But these filaments are in the rock, they have been altered to silica (no reaction to HCL). I have been able to isolate some with freeze thawing a sample. Other portions of the "tufa" have also altered to silica. The colors are so striking, but I think they are different species of algae, cyanobacteria, or bacteria.
I attached a picture of a slab so you can see the stromatolitic (left side) and thrombolitic textures (upper right side)
Reaching back in my memory several decades, I seem to vaguely recall that some types of chlorophyll molecules have been found to be preserved in ancient rocks. I wonder if light transmission spectroscopy could show characteristic absorption bands of one or more types of chlorophylls? (I'm not sure if this would work well in thin-section of rock; minerals could interfere)
More than likely an artifact in sampling. Especially because on further magnification, any definite structure found in algae or in filamentous fungi is not visible.