Perl 5, 87 bytes

sub f{$_=pop;/(.)(.+)/?do{my$s=$1;map s/@_\d+/0+$&/ger,map{("$s $_",$s.$_)}f(1,$2)}:$_}

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Ungolfed:

sub f {
  $_ = pop;                     # set $_ to input (last arg)
  if( /(.)(.+)/ ) {             # if input two or more digits, split
                                # into start digit and rest
    my $s = $1;                 # store start digit
    return
      map s/@_\d+/0+$&/ger,     # no @_ => 1st recursive level => trim leading 0s
                                # 0+$& means int(digits matched)
      map { ("$s $_", "$s$_") } # return "start+space+rest" and "start+rest"...
      f(1, $2)                  # ...for every result of rest
                                # (1 marks recursive level below first)
  }
  else {
    return $_                   # if just one digit, return that
  }
}

Perl 5, 87 bytes

sub f{$_=pop;/(.)(.+)/?do{my$s=$1;map s/@_\d+/0+$&/ger,map{("$s $_",$s.$_)}f(1,$2)}:$_}

Try it online!

Ungolfed:

sub f {
  $_ = pop;                     # set $_ to input (last arg)
  if( /(.)(.+)/ ) {             # if input two or more digits, split
                                # into start digit and rest
    my $s = $1;                 # store start digit
    return
      map s/@_\d+/0+$&/ger,     # no @_ => 1st recursive level => trim leading 0s
                                # 0+$& means int(digits matched)
      map { ("$s $_", "$s$_") } # return "start+space+rest" and "start+rest"...
      f(1, $2)                  # ...for every result of rest
                                # (1 marks recursive level below first)
  }
  else {
    return $_                   # if just one digit, return that
  }
}

Perl 5 -MList::Util, 68 bytes

...which is further golfed from the answer from @xcali

say uniq map"@F
"=~s| |$_/=2;','x($_%2)|reg=~s|\d+|$&*1|reg,1..2**@F

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Perl 5, 87 bytes

sub f{$_=pop;/(.)(.+)/?do{my$s=$1;map s/@_\d+/0+$&/ger,map{("$s $_",$s.$_)}f(1,$2)}:$_}

Try it online!

Perl 5, 87 bytes

sub f{$_=pop;/(.)(.+)/?do{my$s=$1;map s/@_\d+/0+$&/ger,map{("$s $_",$s.$_)}f(1,$2)}:$_}

Try it online!

Ungolfed:

sub f {
  $_ = pop;                     # set $_ to input (last arg)
  if( /(.)(.+)/ ) {             # if input two or more digits, split
                                # into start digit and rest
    my $s = $1;                 # store start digit
    return
      map s/@_\d+/0+$&/ger,     # no @_ => 1st recursive level => trim leading 0s
                                # 0+$& means int(digits matched)
      map { ("$s $_", "$s$_") } # return "start+space+rest" and "start+rest"...
      f(1, $2)                  # ...for every result of rest
                                # (1 marks recursive level below first)
  }
  else {
    return $_                   # if just one digit, return that
  }
}