Thank you John!
Yes, you're right, swords were not widespread by the time the siege of Ancona took place. There are some evidences of course but aren't really many neither are them so common.
The best one is this relief that once decorated the "Porta Romana" in Milan:
It is the best example (as far as I know) of Italian militiamen, representing the soldiers of Milan marching out of the town in order to fight Frederick Redbeard.
Those are the ages the siege of Ancona took place, but this shown here is nevertheless a rich militia as most men wear helmets, some wear mail coats and three (at last) have swords in their hands (plus a falchion).
There are similar reliefs surviving on a church in Ancona:
those are relatively newer than the relief in Milan (approximately 25 to 35 years newer, having been dated to circa 1200) and while 2 out of 4 armed men have swords in their hands, it is not sure they are meant to depict the town's militia.
I haven't been able to find a single iconologic interpretation of the church's portal, but the relief continues with figures of musicians, animals and apostles so I don't really think the armed men represents the town defenders.
I mostly based my reconstruction on a written source, "The siege of Ancona" by Boncompagno da Signa ("Liber de obsidione Ancone", written in 1200).
Altough Boncompagno did not experience the siege himself he claim to have interviewed citizens who fought there and, altough his text is at times full of rhetoric, he still gives us some useful information on the weapons used.
He speaks of this priest who one windy day grab a double-edged axe, undressed himself and went swimming under the biggest Venetian ship (called the "Totus mundus"), chopping its main anchoring rope and therefore allowing the wind to drift the "Totus mundus" against smaller vessels, causing great damages and loss.
Boncompagno also speaks of a woman, Stamira or Stamura, who grabbed an axe and a torch and while the Germans were assaulting the walls she performed a sortie and went to their siege machines, where she broke a barrel containing flammable material which she lit up with the torch, destroying the enemy machines.
He then cites the use of crossbows, spears, bows and shields.
The only thing I'm not really convinced is the flat helmet, which appears to be too shiny new for a militiaman.
But I kind of cheated, by saying that my man represent a wealthy ship's captain or a wealthy merchant or shipowner - all things that allowed him to travel much and put his hands on this sort of innovations earlier than the average fellow citizen
A little bit too articulated but not necessarily unrealistic, in my opinion.
Thank you so much John, it's great to hear from you and a honor to receive your compliments!
(sorry for the many grammar mistakes but it's 3.25 am and I'm too tired to proof read
forgive me)