Instability in DeleteDuplicates and Tally

While preparing an answer to Count how many different values a list takes in Mathematica I came across an instability (for lack of a better term) in both DeleteDuplicates and Tally that I do not understand.

Consider first:

a = {2.2000000000000005, 2.2, 2.1999999999999999};

a // InputForm
DeleteDuplicates@a // InputForm
Union@a // InputForm
Tally@a // InputForm

   {2.2000000000000006`, 2.2, 2.1999999999999997`}

   {2.2000000000000006`, 2.2, 2.1999999999999997`}

   {2.1999999999999997`, 2.2, 2.2000000000000006`}

   {{2.2000000000000006`, 3}}

This behavior is as I expected in each case. Tally compensates for the slight numerical differences and sees each element as being equivalent. Union and DeleteDuplicates see all elements as unique. (This behavior of Tally is not documented to my knowledge, but I have made use of it before.)

Now, consider this complication:

a = {11/5, 2.2000000000000005, 2.2, 2.1999999999999997};

a // InputForm
DeleteDuplicates@a // InputForm
Union@a // InputForm
Tally@a // InputForm

   {11/5, 2.2000000000000006, 2.2, 2.1999999999999997}

   {11/5, 2.2000000000000006, 2.2}

   {2.1999999999999997, 2.2, 11/5, 2.2000000000000006}

   {{11/5, 1}, {2.2000000000000006, 1}, {2.2, 2}}

The output of Union is as anticipated, but the results from both DeleteDuplicates and Tally are surprising.

• Why does DeleteDuplicates suddenly see 2.1999999999999997 as a duplicate to be eliminated?

• Why does Tally suddenly see 2.2000000000000006 and 2.2 as distinct,