When I want to denote a perfect tense like this:
‘she felt very pleased to see the ships have turned around.’
is the auxiliary 'have' supposed to be ‘having’ or ‘have’?
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The sentence as it stands is fine. Your have will be parsed as a finite verb heading the VP of a relative clause with a null-relativizer:
She felt very pleased to see [that] the ships have turned around.
That, however, implies that the turning and the seeing occurred so recently that the ships are still in the turned-around position. I suspect that what you are trying to express is this temporal relationship:
You have a number of options.
She felt very pleased to see that the ships had turned around. - Here the source of her pleasure is not the ships but the fact that they turned around.
She felt very pleased to see the ships having turned around. - This is ambiguous. If having is parsed as a participle (an adjective), the source of her pleasure is the ships in the turned position; but few readers will parse it that way. Most will understand having as a gerund (a nominal), and the reading will be same as (1).
She felt very pleased to see the ships, which had turned around. - This makes the ships the source of her pleasure, but it also makes the new position incidental.
If you want to make it unambiguously clear that the ships in their new position were the source of her pleasure and that the new position was assumed before she saw them, you have to drop the perfect construction and use the bare past participle. In the past context, this is equivalent to a perfect construction:
She felt very pleased to see the turned-around ships.