Hi Bob,
If you are indeed using Pages 6.0, then your OS information for the iMac needs updating. Pages 6.0 requires macOS 10.12 Sierra.
I was able to reproduce your results in Pages 5.6.2 on El Capitan.

New Pages doc.
Set right aligned tabs as shown on ruler.
Typed tab "data" on first line of document, and pressed return.
Selected and copied the text and the tab (omitted the return).
Pressed the down arrow to move to the next line.
Pressed command-V, return five times to create lines 2, 3, 4, 5, 6 with one occurrence of tab "data" on each line.
No space character visible. Possibly due to fact that each of these was a new line.
Clicked to place insertion point between "data" and return on line 1.
Pressed command-V, down arrow twice to paste clipboard contents at ends of lines 1, 2, 3, and move to the end of line 4.
Typed tab, "DATA"
Selected tab, "DATA" (but not return). Copy. Pressed down arrow to move to line 5.
Pressed command-V, down arrow two times to paste clipboard contents at ends of lines 5 and 6 and move to the end of line 4. Leading space added to these two pastes.
Pressed command-V,return three times to add lines 7, 8 and 9, each containing tab "DATA" return (with no visible leading space).
Placed the insertion point between "DATA" and return on line 7.
Typed tab, "data"
Selected tab, "DATA" (but not return). Copy. Pressed down arrow to move to line 8.
Pressed command-V, down arrow two times to paste clipboard contents at ends of lines 8 and 9. Leading space was added to these pastes.
what is happening here is that Pages is 'smart' enough to insert a space to separate one word from another when the insertion point is placed immediately after a word, but is not 'smart' enough to recognize that a tab character is not the start of another word, and that the inserted space is neither needed nor wanted.
Regards,
Barry
Dont know why it happens, but I suggest you determine what to do that consistently produces this behaviour, then file a bug report with Apple via the Provide Pages Feedback item in the Pages menu.
I'll do the same for 5.6.2, with a note that this behaviour has also been reported in v6.0.