> Yes, I can measure a mysterious 1MHz signal
> wherever I like...

    In any digital circuit a 1MHz signal is hardly
    mysterious, rather it is to be expected.

    You mention a tantalum bypass cap without giving
    a value.  Normally a tantalum cap would be at
    least 1uf.  A 'bulk' capacitor like this will
    do nothing to filter high frequency noise.  For
    that you should be using a 0.01uf to 0.1uf
    (ceramic) capacitor.

    You should add this capacitor in parallel to the
    tantalum, and as close to the dac as possible.

___Bob