Pandas merge on multiple columns ignoring NaN

I am trying to do the same as this answer, but with the difference that I'd like to ignore NaN in some cases. For instance:

#df1
     c1    c2    c3
0    a     b     1
1    a     c     2
2    a     nan   1
3    b     nan   3
4    c     d     1
5    d     e     3

#df2
     c1    c2    c4
0    a     nan   1
1    a     c     2
2    a     x     1
3    b     nan   3
4    z     y     2

#merged output based on [c1, c2], dropping instances 
#with `NaN` unless both dataframes have `NaN`.

     c1    c2    c3    c4
0    a     b     1     1   #c1,c2 from df1 because df2 has a nan in c2
1    a     c     2     2   #in both
2    a     x     1     1   #c1,c2 from df2 because df1 has a nan in c2
3    b     nan   3     3   #c1,c2 as found in both
4    c     d     1     nan #from df1
5    d     e     3     nan #from df1
6    z     y     nan   2   #from df2

NaNs may come from either c1 or c2, but for this example I kept it simpler.

I'm not sure what's the cleanest way to do this. I was thinking to merge based on [c1,c2], and then loop by rows with nan, but this will not be so direct. Do you see a better way to do it?

Edit - clarifying conditions
1. No duplicates are found anywhere.
2. No combination is performed between two rows if they both have values. c1 may not be combined with c2, so order must be respected.
3. For the cases where one of the 2 dfs has a nan in either c1 or c2, find the rows in the other dataframe that don't have a full match on both c1+c2, and use it. For instance:

• (a,c) has a match in both so it is no longer discussed.
• (a,b) is only in df1. No b is found in df2.c2. The only row in df2 with a known key and a nan is row 0 so it is combined with this one. Note that order must be respected this is why (a,b) #df1 cannot be combined with any other row of df2 that also contains a b.
• (a,x) is only in df2. No x is found in df1.c2. The only row in df1 with one of the known keys with a nan is row with index 2.