Draw Translucent Boxes
Python 3, 289 286 bytes



l,u=eval(input())
*_,w,h=map(max,zip(*l))
r=[*map(list,[' '*-~w]*-~h)]
R=range
for x,y,X,Y in l:
 for i in R(x,X+1):
  for j in R(y,Y+1):Q=i&lt;X,y&lt;j,x&lt;i,j&lt;Y;r[j][i]=u[sum(3**o*max(Q[o]*[j in(y,Y),i in(x,X)][o%2]*2,u.index(r[j][i])//3**o%3-Q[o])for o in R(4))]
for l in r:print(''.join(l))


Try it online!

Takes input as a list of 4-tuples: (x0, y0, x1, y1), along with the pipedrawing characters as follows: " ╶╺╵└┕╹┖┗╴─╼┘┴┶┚┸┺╸╾━┙┵┷┛┹┻╷┌┍│├┝╿┞┡┐┬┮┤┼┾┦╀╄┑┭┯┥┽┿┩╃╇╻┎┏╽┟┢┃┠┣┒┰┲┧╁╆┨╂╊┓┱┳┪╅╈┫╉╋"

Supports boxes of zero width or height (uses all box drawing chars).

Based on my answer to this question, but modified to alter existing boxes when drawing.

'Ungolfed':

u=" ╶╺╵└┕╹┖┗╴─╼┘┴┶┚┸┺╸╾━┙┵┷┛┹┻╷┌┍│├┝╿┞┡┐┬┮┤┼┾┦╀╄┑┭┯┥┽┿┩╃╇╻┎┏╽┟┢┃┠┣┒┰┲┧╁╆┨╂╊┓┱┳┪╅╈┫╉╋"

#Create array of spaces:
l=eval(input())
w,h=list(map(max,zip(*l)))[2:]
r=[[' ']*w for _ in' '*h]

for x,y,X,Y in l:
 n,m=X-1,Y-1
 for i in range(x,X):
  for j in range(y,Y):
   A,B=j in(y,m),i in(x,n)
   P=(i&lt;n*A,y&lt;j*B,x&lt;i*A,j&lt;m*B)   # Pipe sides of the new box
   Q=(i&lt;n,y&lt;j,x&lt;i,j&lt;m)           # Pipe sides that are inside the new box, and should be thinned.
   # Get new pipe (sides of new box, on top of thinned pipes of existing boxes. (u.index... is existing pipe))
   l=sum(3**o*max(P[o]*2,(u.index(r[j][i])//3**o%3)-Q[o])for o in range(4))
   #Add to 'canvas'
   r[j][i]=u[l]

print('\n'.join(''.join(l)for l in r))