Visualize the Euclidean algorithm

JavaScript (ES6), 128 124 bytes

t=0
f=
(a,b,o,c,d)=>setInterval(e=>{e=[b,d,a,c];o.data=`-0
-0`.replace(/./g,c=>c.repeat(e.pop()));c|d?c=d=0:a>b?a-=c=b:b-=d=a},1e3)

<form><input id=a><input id=b><button onclick=clearTimeout(t),t=f(+a.value,+b.value,o.firstChild)>Go!</button><pre id=o>

Jelly, 29 bytes

VY“ñc‘ỌœS.⁸
1ẋǵ+M¦ṚÇt€2ǵ⁻/¿

Try it online!

This defines a function 2Ŀ (not a full program; the TIO link contains a footer that converts a function into a program) that takes a list of two elements as input, and displays the output on the screen (one of our legal I/O methods, and one that is kind-of necessary for this challenge because it talks about the appearance on screen). This assumes that the program is run in a terminal that complies with the ANSI standard (I used gnome-terminal but most will work), and that the terminal is initially empty (which seems like the most sensible default); note that Try it online! does not conform to these assumptions, and thus the output is distorted there (I ran the program locally to verify that it animates as expected). I use 1 where the question uses 0, and 2 in place of -.

Explanation

Helper function 1Ŀ (given a list of two lists of digits, outputs them on the first and second lines of the screen, then waits 0.5 seconds; returns its input)

VY“ñc‘ỌœS.⁸
V                   Convert each list of digits to an integer
 Y                  Separate these integers by newlines
  “ñc‘              {Output that; then restart with} the list [27, 99]
      Ọ             Convert codepoints to characters (i.e. "\x1bc"
       œS.          Wait (œS) 0.5 (.) seconds
          ⁸         {Output that; then return} the initial argument

The string "\x1bc", when sent to an ANSI-compatible terminal, is interpreted as a control code to reset the terminal; this clears the screen and moves the cursor to the top left corner (thus resetting the terminal ready for the next output).

The helper function is named 1Ŀ (Jelly autogenerates names of this form for functions, and in fact there's no other way to name them), but it can be referred to simply as Ç from the main program (because the language has shorthand for functions with numbers nearby).

Main function 2Ŀ (implements the task requested in the question)

1ẋǵ+M¦ṚÇt€2ǵ⁻/¿
1ẋ                  Convert input to unary
  Ç                 Call helper function (producing one animation frame)
   µ         µ  ¿   While
              ⁻/      the elements differ:
     M¦               Change the largest element
    +  Ṛ                by adding corresponding elements of the other element
        Ç             Call helper function (producing one animation frame)
         t€2          Delete all 2s from each side of each element
            Ç         Call helper function (producing one animation frame)

Python 2, 152 146 bytes

import time
s,o='-0'
a,b=input()
while a*b:
 d,e=o*a,o*b
 if a>b:a=a-b;d=s*b+o*a
 elif b>a:b=b-a;e=s*a+o*b
 else:a=0
 print d+'\n'+e;time.sleep(1)

Try it online!

Takes two comma-separated integers as input