How about a countably infinite string of machines similar to my original
machines but having no input where at each print cycle each machine prints
a one or zero at random.
The stings they produce read left to right or right to left is a number = a
universe configuration. Eventually some selection of output numbers will
match the sequence of numbers for any universe. A particular universe
would not necessarily get its next number right away - it would usually
wait a random number of print cycles to get to its next number while some
other universes get their next number.
That seems to work.