#JavaScript (ES6), 377376 416 494
Just to be there ...
This build all the tilings, then choose a random one. Time for the 232848 tilings for N=4 is ~45 sec on my laptop. I did not try N=5.
Being EcmaScript 6 it runs on Firefox only.
F=n=>{
for(i=s=r=b='';i<n;i++)
s='\n'+b+'/\\'[R='repeat'](n-i)+'_\\'[R](n)+s,
r+='\n'+b+'\\/'[R](n-i)+'_/'[R](n),
b+=' '';
for(h=[t=0],x=[s+r];s=x[t++];x=[s+r];s=x[t];t++)
for(d='';!d[n*3];d+='.')
if(l=s.match(r=RegExp("(\n"+d+"/)\\\\_(.*\n"+d+"\\\\)/_","g")))
for(j=2<<l.length;j-=2;h[z]||(h[z]=x.push(z)))
z=s.replace(r,(r,g,h)=>(k+=k)&j?g+'_/'+h+'_\\':r,k=1);
return b+'__'[R](n)+x[Math.random()*(t-1)|0]*t|0]
}
function go()
{
var n = N.value | 0,
t0 = performance.now(),
r = F(n),
t1 = performance.now();
O.innerHTML = r+'\n\nTime (msec): '+(t1-t0)
}
N: <input id=N value=3><button onclick="go()">GO</button>
<pre id=O></pre>