Write a Python function `check_equal_tuple_lengths(tuples_list)` that determines whether all tuples in a given list have the same length. The function should return `True` if all tuples have the same length, and `False` otherwise.
#### Example Usage
```python [main.nopy]
check_equal_tuple_lengths([(1, 2, 3), (4, 5, 6)]) # Returns: True
check_equal_tuple_lengths([(1, 2), (3, 4, 5)]) # Returns: False
check_equal_tuple_lengths([]) # Returns: True
```
#### Constraints
- The input will always be a list of tuples.
- The function should handle an empty list gracefully.