```ts
/**
 * Replay tile requests against a size-limited cache of fresh renders.
 */
export function tileRenders(requests: [number, string][], freshFor: number, size: number): [number, string][] {
  if (!Number.isInteger(freshFor) || freshFor < 1 || !Number.isInteger(size) || size < 1) {
    throw new Error("freshFor and size must be positive integers");
  }
  
  const held = new Map<string, number>();
  const renders: [number, string][] = [];
  
  for (const [tick, name] of requests) {
    // Remove tiles that are no longer fresh
    for (const [heldName, heldTick] of held.entries()) {
      if (tick >= heldTick + freshFor) {
        held.delete(heldName);
      }
    }

    // Serve a still-fresh render if available
    if (held.has(name)) {
      renders.push([tick, name]);
      continue;
    }

    // If the cache is full, drop the oldest tile
    if (held.size >= size) {
      const oldest = Array.from(held.entries()).sort((a, b) => a[1] - b[1])[0];
      held.delete(oldest[0]);
    }

    // Record the new render
    held.set(name, tick);
    renders.push([tick, name]);
  }
  
  return renders;
}
```