turing machine for the language L ={w#w' where w<w'}
Here is a sketch of a construction method of the TM:


Compare bit by bit from the first bit to the last bit as normal. If equals, proceed to the next bit. Otherwise, it can decide the order.
If the head of TM reaches the word separator $\#$ or the blank symbol (say $\$$ or something), finish the comparison. If both words end simultaneously (i.e. when $\left| w \right| = \left| w' \right|$), then $w = w'$. If not, the shorter word is less than the other in the lexicographic order.


This idea is the same for both a deterministic TM and a non-deterministic TM. Only a non-deterministic TM can compare bits in a non-deterministic way.