Brain-Flak, 175 bytes (174 chars + 1 flag)

Run with -c flag.

{(({})<(())>){({}[()]<([{}])>)}{}(({}<>{}<><({}<>)((()()()()()){})>)<{({}[()]<((((()()()()){}){}){})>)}>{})<>}<>{}{({}<>)(({})(())){({}[()]<([{}]())>)}{}{<>{}<>(<{}>)}{}<>}<>

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##Explanation

{ for each char in the input...
  (({})<(())>){({}[()]<([{}])>)}{} push 1/-1 for backslash/slash
  ((
   {}<>{}<> add the 1/-1 to a running total
   <
    ({}<>) move slash/backslash to other stack
    ((()()()()()){}) newline
   >
  )<{({}[()]<((((()()()()){}){}){})>)}>{}) spaces
  <>
}<>{} end for
reverse data order, removing one space before backslash
{({}<>)(({})(())){({}[()]<([{}]())>)}{}{<>{}<>(<{}>)}{}<>}<>

Brain-Flak, 175 bytes (174 chars + 1 flag)

Run with -c flag.

{(({})<(())>){({}[()]<([{}])>)}{}(({}<>{}<><({}<>)((()()()()()){})>)<{({}[()]<((((()()()()){}){}){})>)}>{})<>}<>{}{({}<>)(({})(())){({}[()]<([{}]())>)}{}{<>{}<>(<{}>)}{}<>}<>

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Explanation

{ for each char in the input...
  (({})<(())>){({}[()]<([{}])>)}{} push 1/-1 for backslash/slash
  ((
   {}<>{}<> add the 1/-1 to a running total
   <
    ({}<>) move slash/backslash to other stack
    ((()()()()()){}) newline
   >
  )<{({}[()]<((((()()()()){}){}){})>)}>{}) spaces
  <>
}<>{} end for
reverse data order, removing one space before backslash
{({}<>)(({})(())){({}[()]<([{}]())>)}{}{<>{}<>(<{}>)}{}<>}<>

Brain-Flak, 175 bytes (174 chars + 1 flag)

Run with -c flag.

{(({})<(())>){({}[()]<([{}])>)}{}(({}<>{}<><({}<>)((()()()()()){})>)<{({}[()]<((((()()()()){}){}){})>)}>{})<>}<>{}{({}<>)(({})(())){({}[()]<([{}]())>)}{}{<>{}<>(<{}>)}{}<>}<>

Try it online!

##Explanation

{ for each char in the input...
  (({})<(())>){({}[()]<([{}])>)}{} push 1/-1 for backslash/slash
  ((
   {}<>{}<> add the 1/-1 to a running total
   <
    ({}<>) move slash/backslash to other stack
    ((()()()()()){}) newline
   >
  )<{({}[()]<((((()()()()){}){}){})>)}>{}) spaces
  <>
}<>{} end for
reverse data order, removing spaces before backslashs
{({}<>)(({})(())){({}[()]<([{}]())>)}{}{<>{}<>(<{}>)}{}<>}<>

Brain-Flak, 175 bytes (174 chars + 1 flag)

Run with -c flag.

{(({})<(())>){({}[()]<([{}])>)}{}(({}<>{}<><({}<>)((()()()()()){})>)<{({}[()]<((((()()()()){}){}){})>)}>{})<>}<>{}{({}<>)(({})(())){({}[()]<([{}]())>)}{}{<>{}<>(<{}>)}{}<>}<>

Try it online!

##Explanation

{ for each char in the input...
  (({})<(())>){({}[()]<([{}])>)}{} push 1/-1 for backslash/slash
  ((
   {}<>{}<> add the 1/-1 to a running total
   <
    ({}<>) move slash/backslash to other stack
    ((()()()()()){}) newline
   >
  )<{({}[()]<((((()()()()){}){}){})>)}>{}) spaces
  <>
}<>{} end for
reverse data order, removing one space before backslash
{({}<>)(({})(())){({}[()]<([{}]())>)}{}{<>{}<>(<{}>)}{}<>}<>