Manually sorting posts within each category

I’m wondering if there is a way (natively or through a plugin/re-coding) to allow a post to appear in more than one category, but ALSO have its own position in each of the categories.

for example…

Four different posts:
Post 1
Post 2
Post 3
Post 4

Three categories:
Cat 1
Cat 2
Cat 3

Cat 1:
Post 2
Post 1
Post 3
(In this category post 2 is in position one, post 1 is position two, post 3 is in position three)

Cat 2:
Post 4
Post 3
(In this category post 4 is in position one and post 3 is position two. Post 3 is also found in Cat 1)

I can give each post a custom field called ‘position’ and sort the category by that value, but how can I make that field different for each category…??