Shuffling a list while keeping order relative to related elements

I'm looking to shuffle a list of the elements $a_1,\dots, a_6, \dots, e_1, \dots, e_6$

while keeping two rules:

if I loop though the list and filter out a specific letter or number it should be in order:

$a_1, a_2, a_3 \dots$ or $a_1, b_1, c_1 \dots$

How can I shuffle the list keeping these rules? I'm using python, so if there's a library that'd be great. Otherwise, just a generic way I could tackle this problem.

Here's an example of a shuffle that would fit the criteria:

$a_1, b_1, a_2, b_2, c_1, a_3, d_1, c_2, d_2, e_1, a_4, b_3,$ $c_3, d_3, b_4, d_4, c_4, a_5, e_2, d_5, e_3, c_5, a_6, b_5, e_4, a_7, $ $b_6, c_6, b_7, d_6, e_5, e_6, c_7, d_7, e_7$

Your problem can be phrased as generating a random linear extension of a partial order. The partial order in your phrase is generated by your constraints. There is a classical algorithm of Matthews described in his paper Generating a random linear extension of a partial order. This might even be implemented in some library.

Your particular case is generating a random lattice word of certain content (for the reduction, delete all letters or all numbers). This is a concept occurring in representation theory of the symmetric groups, generalizing the classical ballot sequences. If you're lucky you might find an algorithm for generating a random lattice word (in Mupad, for example, somebody implemented an algorithm for generating all lattice words).

StackExchange Computer Science Q#80108, answer score: 8