Floaties in the water can result from different circumstances. The main drawback to having filtrated water is that many good compounds, such as Chlorine, are removed along with impurities. This leaves the water in the holding reservoir subject to biologic processes, including algae and other organisms. One thing you can do is flush the tank for about 2 minutes each week to help pass residuals thru the line.
In some cases, certain types of water lines can induce taste and cosmetic appearance of the water. I have found that most of the soft, vinyl blue and white water lines cause taste and floaties after awhile. I found this only after responding to numerous complaints while at Factory Service. After replacing those water lines with copper or stainless hoses, the issues went away.
Ultimately, you may have to replace the water reservoir to start anew, however the floaties are not pathogenic so they will only cause you cosmetic discomfort.