Make a group_indices based on several columns

I would like to generate indices to group observations based on two columns. But I want groups to be made of observation that share, at least one observation in commons. I can see how to make groups based on observations that share both observation in common, but not just one of them.

For example, with the data frame :

dt <- data.frame(id=1:10,
             G1 = c("A","A","B","B","C","C","C","D","E","F"),
             G2 = c("Z","X","X","Y","W","V","U","s","T","T"))

I would like to get a column

1,1,1,1,2,2,2,3,4,4

I tried with group_indices from dplyr, but haven't managed it.