df <- data.frame('a'=c('a, b, c, d', 'a, c', 'b, d'), 'b'=c('a, d', 'a', 'a, d'))

           a      b
1 a, b, c, d   a, d
2       a, c      a
3       b, d   a, d

a      b      c
1 a, b, c, d   a, d   a, d
2       a, c      a      a
3       b, d   a, d      d

df <- df %>%
    mutate(c=paste(c(intersect(unlist(strsplit(a, split=", ")), unlist(strsplit(b, split=", "))))))

Error in eval(substitute(expr), envir, enclos) :
  wrong result size (2), expected 3 or 1

df %>%
    mutate(c=length(c(intersect(unlist(strsplit(a, split=", ")), unlist(strsplit(b, split=", "))))))

           a    b   c
1 a, b, c, d a, d   2
2       a, c    a   2
3       b, d a, d   2

df %>% mutate(a_list=list(unlist(strsplit(a, split=", "))))

Error in eval(substitute(expr), envir, enclos) :
   not compatible with STRSXP