What is the role of 清楚 in 一定要注意签订书面合同，核实清楚物业费、水费、...? What is the role of 清楚 in this sentence? Could they have written '核实得清楚‘ = 'verify clearly' or does ‘清楚‘ somehow work with '拖欠‘ = verify (the property) is clear of arrears.

I suppose it doesn't affect the meaning much either way. I just had a feeling it might not be a straightforward '核实（得）清楚‘。

The choices I see are:

核实清楚 verify clearly

清楚拖欠 be clear of arrears 


  女：在签订二手房买卖合同时，一定要注意签订书面合同，核实清楚物业费、水费、煤气费‘电话费等与房屋有关的费用是否存在拖欠，在合同中要注明卖方迁出户口的期限，要注意保留好收据。
清楚 here is resultant complement for 核实 instead of adverbial, so the literal meaning is 'verify clear' (verify towards a clear result, think 'make clear') instead of 'verify clearly'.

If you have to insert 的/地/得, it should be 得 i.e. 核实得清楚, but that changes the sentence from imperative to declarative.

It's really just a figure of speech to emphasize 核实, as opposed to carry some actual meaning. The previous sentence '一定要注意...' is similar in that regard, by using 一定 to emphasize 注意.