Almost all iron filters operate under the principle of precipitation as there are no inexpensive ion exchange methods. Greensand is a type of manganese containing mineral that acts as a catalyst to help promote the oxidation of iron from its soluble +2 ion form to the largely insoluble +3 ion (rust is iron oxide with the iron in a +3 oxidation state). Introduction of an oxidant (chlorine or potassium permanganate) helps to regenerate the greensand. Sometimes air is introduced through a Venturi inlet to help promote the oxidation of iron water into its insoluble form. Once the iron precipitates, it can be removed by sand filtration (the final portion of the greensand filter is basically a sand filter). The efficiency of the iron removal is highly dependent on many factors (initial iron concentration, water flow rate, condition of the manganese sand, dissolved oxygen content, etc) and and it is very easy to overwhelm an iron filter if conditions are not properly calibrated.
I suspect what is going on in the OP’s tub is simply the result of a small amount of iron making it past the household filter system. It’s not enough iron to cause issues in the house (water taste, stained toilet bowls, etc) but it’s enough for chlorine to react with and cause the yellow/brown tea colored water.