And that's how you can explain mysteriously frequent consecutive runs and alternations. If the last two bits being 01 (respectively 10) makes it more likely for the next bit to be 0 (respectively 1), and the last two bits being 00 (respectively 11) makes it more likely for the next bit to be 0 (respectively 1), then you would be more likely to see both long 0000... or 1111... consecutive runs and 01010... alternations.

Alternatively, it could be a first order chain that checks the digit located 2 bits before it and makes it more likely to match. This does however remove the possibility of assigning different likelihoods to 00  and 01 being followed by 0 (or 11 and 10 being followed by 1).

Edit: or a coin biased to match the digit 2 entries before, removing the possibility of different likelihoods for chains of 1 and chains of 0 (or alternating chains beginning/ending on 1 or 0).